1806:
8530:
4901:
752:
1505:
4664:
3793:
2572:
887:
4660:
1801:{\displaystyle {\begin{aligned}{\mathcal {L}}_{\text{SG}}(\varphi )&={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-{\frac {\varphi ^{2}}{2}}+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}\\&={\mathcal {L}}_{\text{KG}}(\varphi )+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}.\end{aligned}}}
1054:
6103:
4896:{\displaystyle A_{v}={\begin{pmatrix}-{\frac {i}{4\lambda }}\cos \varphi &-{\frac {1}{4\lambda }}\sin \varphi \\{\frac {1}{4\lambda }}\sin \varphi &{\frac {i}{4\lambda }}\cos \varphi \end{pmatrix}}=-{\frac {1}{4\lambda }}i\sin \varphi \sigma _{2}-{\frac {1}{4\lambda }}i\cos \varphi \sigma _{3},}
794:
There are also some more straightforward ways to construct new solutions but which do not give new surfaces. Since the sine-Gordon equation is odd, the negative of any solution is another solution. However this does not give a new surface, as the sign-change comes down to a choice of direction for
748:, that if a pair of matrix-valued bilinear forms satisfy the GaussâCodazzi equations, then they are the first and second fundamental forms of an embedded surface in 3-dimensional space. Solutions to the sine-Gordon equation can be used to construct such matrices by using the forms obtained above.
3612:
4289:
where a constant energy density has been added so that the potential is non-negative. With it the first two terms in the Taylor expansion of the potential coincide with the potential of a massive scalar field, as mentioned in the naming section; the higher order terms can be thought of as
2446:
3150:
2950:
2797:
4498:
3359:
2631:
relating the transformed results. The 2-soliton solutions of the sine-Gordon equation show some of the characteristic features of the solitons. The traveling sine-Gordon kinks and/or antikinks pass through each other as if perfectly permeable, and the only observed effect is a
3211:
3-soliton collisions between a traveling kink and a standing breather or a traveling antikink and a standing breather results in a phase shift of the standing breather. In the process of collision between a moving kink and a standing breather, the shift of the breather
4287:
1368:
941:
2328:
1935:
5407:
5956:
4129:
3010:
1490:
2804:
5765:. This is sometimes known as the Coleman correspondence and serves as an example of boson-fermion correspondence in the interacting case. This article also showed that the constants appearing in the model behave nicely under
2654:
6570:
3788:{\displaystyle {\begin{aligned}\psi _{u}&=\varphi _{u}+2a\sin {\Bigl (}{\frac {\psi +\varphi }{2}}{\Bigr )}\\\psi _{v}&=-\varphi _{v}+{\frac {2}{a}}\sin {\Bigl (}{\frac {\psi -\varphi }{2}}{\Bigr )}\end{aligned}}\,\!}
5709:
One can also consider the sine-Gordon model on a circle, on a line segment, or on a half line. It is possible to find boundary conditions which preserve the integrability of the model. On a half line the spectrum contains
4293:
The topological charge is conserved if the energy is finite. The topological charge does not determine the solution, even up to
Lorentz boosts. Both the trivial solution and the soliton-antisoliton pair solution have
4158:
3245:
2079:
2336:
614:
402:
308:
5612:
5303:
5482:
1168:
5030:
1241:
4655:{\displaystyle A_{u}={\begin{pmatrix}i\lambda &{\frac {i}{2}}\varphi _{u}\\{\frac {i}{2}}\varphi _{u}&-i\lambda \end{pmatrix}}={\frac {1}{2}}\varphi _{u}i\sigma _{1}+\lambda i\sigma _{3},}
3617:
1510:
898:
Consider a line of pendula, hanging on a straight line, in constant gravity. Connect the bobs of the pendula together by a string in constant tension. Let the angle of the pendulum at location
1999:
6680:
6155:
3601:
1260:
6298:
720:
453:
5699:
5074:
3476:
6453:, which are respectively the free fermion point, as the theory is dual to a free fermion via the Coleman correspondence, and the self-dual point, where the vertex operators form an
5934:
2532:
6339:
2492:
671:
5655:
2213:
6375:
6245:
4369:
5800:
3237:
3993:
8925:
6602:
6451:
6415:
4466:
4402:
3912:
1170:. However this does give an intuitive picture for the sine-gordon equation. One can produce exact mechanical realizations of the sine-gordon equation by more complex methods.
845:
2609:
2181:
1090:
1049:{\displaystyle \underbrace {m\varphi _{tt}} _{\text{mass times acceleration}}=\underbrace {T\varphi _{xx}} _{\text{tension}}-\underbrace {mg\sin \varphi } _{\text{gravity}}}
5162:
3389:
5881:
5854:
5827:
4928:
6194:
2152:
6487:
7960:
6949:
6098:{\displaystyle {\mathcal {L}}_{QsG}={\frac {1}{2}}\partial _{\mu }\varphi \partial ^{\mu }\varphi +{\frac {1}{2}}m_{0}^{2}\varphi ^{2}-\alpha (V_{\beta }+V_{-\beta })}
5505:
4493:
4153:
3988:
3886:
3840:
3552:
2126:
1838:
936:
813:
510:
124:
56:
5321:
3409:
2233:
2102:
5901:
5219:
5192:
4437:
3932:
3860:
3820:
3436:
238:
6622:
4318:
722:
Thus, any pseudospherical surface gives rise to a solution of the sine-Gordon equation, although with some caveats: if the surface is complete, it is necessarily
2454:
The 1-soliton solutions can be visualized with the use of the elastic ribbon sine-Gordon model introduced by Julio
Rubinstein in 1970. Here we take a clockwise (
8383:
8193:
3952:
916:
865:
96:
76:
1387:
8564:
8275:
8234:
2249:
727:
6690:
A supersymmetric extension of the sine-Gordon model also exists. Integrability preserving boundary conditions for this extension can be found as well.
8913:
8013:
7645:
Inami, Takeo; Odake, Satoru; Zhang, Yao-Zhong (1995). "Supersymmetric extension of the sine-Gordon theory with integrable boundary interactions".
7455:
Zamolodchikov, Alexander B.; Zamolodchikov, Alexey B. (1978). "Relativistic factorized S-matrix in two dimensions having O(N) isotopic symmetry".
3145:{\displaystyle \varphi (x,t)=4\arctan \left({\frac {{\sqrt {1-\omega ^{2}}}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2}}}\;x)}}\right).}
35:
5549:
8038:
8142:
4937:
8644:
8373:
7709:
7171:
7146:
7103:
6984:
6907:
6882:
2945:{\displaystyle \varphi _{K/AK}(x,t)=4\arctan \left({\frac {v\cosh {\frac {x}{\sqrt {1-v^{2}}}}}{\sinh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)}
7536:
Fröb, Markus B.; Cadamuro, Daniela (2022). "Local operators in the Sine-Gordon model: $ \partial_ÎŒÏ\, \partial_ÎœÏ$ and the stress tensor".
2792:{\displaystyle \varphi _{K/K}(x,t)=4\arctan \left({\frac {v\sinh {\frac {x}{\sqrt {1-v^{2}}}}}{\cosh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)}
8265:
7953:
8782:
3354:{\displaystyle \Delta _{\text{B}}={\frac {2\operatorname {artanh} {\sqrt {(1-\omega ^{2})(1-v_{\text{K}}^{2})}}}{\sqrt {1-\omega ^{2}}}},}
8121:
6709:
are well-described by the sine-Gordon equations, and conversely provide a useful experimental system for studying the sine-Gordon model.
5225:
for the sine-Gordon equation, in the sense that the zero-curvature equation recovers the PDE rather than them satisfying Lax's equation.
2441:{\displaystyle \varphi '_{v}=-\varphi _{v}+{\frac {2}{\beta }}\sin {\frac {\varphi '-\varphi }{2}}{\text{ with }}\varphi =\varphi _{0}=0}
2215:
are known as vacuum states, as they are constant solutions of zero energy. The 1-soliton solution in which we take the negative root for
8003:
7928:
6733:
2010:
458:
This is the original form of the sine-Gordon equation, as it was considered in the 19th century in the course of investigation of
8697:
3504:
3486:
3192:
3173:
3159:
2972:
2959:
2556:
2542:
9019:
8533:
8213:
6300:, where a finite number of counterterms are needed to render the theory well-posed. More counterterms are needed for each threshold
459:
8767:
890:
A line of pendula, with a "breather pattern" oscillating in the middle. Unfortunately, the picture is drawn with gravity pointing
680:
525:
335:
246:
9214:
8557:
8463:
8270:
8033:
5247:
8066:
5515:. This is no longer a soliton equation, but it has many similar properties, as it is related to the sine-Gordon equation by the
5426:
8423:
8326:
8306:
7946:
1095:
8727:
9204:
8229:
7988:
7983:
4282:{\displaystyle E=\int _{\mathbb {R} }dx\left({\frac {1}{2}}(\varphi _{t}^{2}+\varphi _{x}^{2})+m^{2}(1-\cos \varphi )\right)}
1188:
737:, also known as the tractroid, corresponds to a static one-soliton, but the tractroid has a singular cusp at its equator.
626:
9224:
9219:
8687:
8599:
8378:
6454:
5658:
3865:
By using a matrix system, it is also possible to find a linear BĂ€cklund transform for solutions of sine-Gordon equation.
767:
The study of this equation and of the associated transformations of pseudospherical surfaces in the 19th century by
8826:
8502:
8028:
6699:
5309:
1946:
1363:{\displaystyle {\mathcal {L}}_{\text{SG}}(\varphi )={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-1+\cos \varphi .}
1247:
876:
162:
8717:
8550:
8497:
8291:
7385:
Bogoliubov, N. M.; Korepin, V. E.; Izergin, A. G. (1985). "Structure of the vacuum in the quantum sine-Gordon model".
6631:
6111:
8071:
5734:. The number of the breathers depends on the value of the parameter. Multiparticle production cancels on mass shell.
674:
142:
3560:
751:
8798:
7769:"Classical and quantum statistical mechanics in one and two dimensions: Two-component Yukawa â and Coulomb systems"
7701:
The sine-Gordon Model and its
Applications: From Pendula and Josephson Junctions to Gravity and High-Energy Physics
7557:
Chandra, Ajay; Hairer, Martin; Shen, Hao (2018). "The dynamical sine-Gordon model in the full subcritical regime".
7095:
6481:
and Hao Shen allowing heuristic results from the quantum sine-Gordon theory to be proven in a statistical setting.
776:
8239:
6258:
5076:. The zero-curvature equation is so named as it corresponds to the curvature being equal to zero if it is defined
1179:
413:
9061:
8919:
8666:
6835:
Hirota, Ryogo (November 1972). "Exact
Solution of the Sine-Gordon Equation for Multiple Collisions of Solitons".
5663:
5313:
5037:
2986:
Another interesting 2-soliton solutions arise from the possibility of coupled kink-antikink behaviour known as a
772:
9024:
8842:
8992:
8475:
6744:
8347:
8172:
8147:
3441:
740:
Conversely, one can start with a solution to the sine-Gordon equation to obtain a pseudosphere uniquely up to
1059:
Note that this is not exactly correct, since the net force on a pendulum due to the tension is not precisely
9138:
9086:
8587:
8203:
8086:
6706:
5906:
5750:
3822:
which will also satisfy the sine-Gordon equation. This is an example of an auto-BĂ€cklund transform, as both
2501:
620:
470:
6303:
2461:
8897:
8882:
8847:
8692:
8514:
8438:
8433:
8368:
8296:
8198:
8101:
8091:
8076:
7993:
6743:
The sine-Gordon equation also arises as the formal continuum limit of a different model of magnetism, the
5626:
5542:
2186:
516:
314:
8772:
8249:
8152:
8116:
8061:
6344:
6214:
5621:
4338:
9076:
8487:
8482:
8321:
8316:
8106:
6943:
6737:
5772:
5516:
3215:
3529:
2624:
2240:
7187:
Yuanxi, Xie; Tang, Jiashi (February 2006). "A unified method for solving sinh-Gordonâtype equations".
6575:
6420:
6384:
5730:. The particle spectrum consists of a soliton, an anti-soliton and a finite (possibly zero) number of
4442:
4378:
3891:
9143:
9112:
8428:
7998:
7969:
7864:
7827:
7780:
7741:
7664:
7601:
7503:
7464:
7429:
7394:
7359:
7350:
Takada, Satoshi; Misawa, Susumu (1981). "The
Quantum Sine-Gordon Model and the Fermi-Bose Relation".
7316:
7273:
7196:
7014:
6844:
5723:
4372:
2628:
818:
741:
7732:
JosĂ©, Jorge (15 November 1976). "Sine-Gordon theory and the classical two-dimensional x â y model".
2578:
2157:
1062:
9102:
8819:
8622:
8492:
8458:
8393:
8311:
8301:
8096:
7698:
Mazo, Juan J.; Ustinov, Alexey V. (2014). "The sine-Gordon
Equation in Josephson-Junction Arrays".
6965:"The sine-Gordon Model: General Background, Physical Motivations, Inverse Scattering, and Solitons"
6963:
Malomed, Boris A. (2014), Cuevas-Maraver, JesĂșs; Kevrekidis, Panayotis G.; Williams, Floyd (eds.),
5079:
1496:
158:
150:
6201:
6200:
properties of the sine-Gordon theory change. The identification of these regimes is attributed to
3367:
9071:
8969:
8964:
8959:
8877:
8809:
8126:
7891:
7796:
7680:
7654:
7617:
7591:
7558:
7537:
7491:
7332:
7289:
7263:
6729:
6713:
6461:, and the theory becomes strictly renormalizable (renormalizable, but not super-renormalizable).
5859:
5832:
5805:
4906:
2243:
to the trivial (vacuum) solution and the integration of the resulting first-order differentials:
1930:{\displaystyle \varphi _{\text{soliton}}(x,t):=4\arctan \left(e^{m\gamma (x-vt)+\delta }\right),}
1251:
463:
181:
146:
7699:
6172:
5402:{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-\cosh \varphi .}
2131:
476:
Consider an arbitrary pseudospherical surface. Across every point on the surface there are two
9209:
9199:
8892:
8573:
8177:
7705:
7579:
7167:
7142:
7099:
6980:
6903:
6878:
6798:
5746:
5617:
5490:
4471:
4138:
3973:
3871:
3825:
3537:
2645:
2111:
921:
798:
756:
495:
173:
109:
41:
7855:
Kogut, John B. (1 October 1979). "An introduction to lattice gauge theory and spin systems".
3394:
2218:
2087:
480:. This allows us to construct a distinguished coordinate system for such a surface, in which
9081:
8654:
8637:
8448:
7872:
7835:
7788:
7749:
7672:
7609:
7511:
7472:
7437:
7402:
7367:
7324:
7281:
7231:
7204:
7070:
7022:
6972:
6924:
6852:
6756:
6717:
6628:. The space dimension is fixed to 2. In the proof of existence of solutions, the thresholds
5886:
5742:
477:
7307:
Korepin, V. E. (1979). "Direct calculation of the S matrix in the massive
Thirring model".
7254:
Bowcock, Peter; Tzamtzis, Georgios (2007). "The complex sine-Gordon model on a half line".
7059:"Neuronic system inside neurons: molecular biology and biophysics of neuronal microtubules"
6875:
Solitons and
Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory
5197:
5170:
4410:
3917:
3845:
3805:
3414:
488: = constant are the asymptotic lines, and the coordinates are incremented by the
211:
9169:
9051:
8852:
8470:
8208:
8157:
7932:
7919:
7913:
7123:
6607:
6197:
6158:
5948:
5766:
5727:
4124:{\displaystyle N={\frac {1}{2\pi }}\int _{\mathbb {R} }d\varphi ={\frac {1}{2\pi }}\left.}
2495:
2455:
780:
103:
7925:
4297:
1485:{\displaystyle \cos(\varphi )=\sum _{n=0}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}},}
7868:
7831:
7784:
7745:
7668:
7605:
7507:
7468:
7433:
7398:
7363:
7320:
7285:
7277:
7200:
7092:
BĂ€cklund and
Darboux Transformations: Geometry and Modern Applications in Soliton Theory
7018:
6964:
6848:
9174:
9164:
9107:
9056:
8954:
8940:
8887:
8777:
8671:
8632:
8413:
8388:
8167:
6725:
5941:
5762:
5758:
5738:
4931:
4332:
3937:
2323:{\displaystyle \varphi '_{u}=\varphi _{u}+2\beta \sin {\frac {\varphi '+\varphi }{2}},}
938:, then schematically, the dynamics of the line of pendulum follows Newton's second law:
901:
850:
723:
81:
61:
795:
the normal to the surface. New solutions can be found by translating the solution: if
9193:
9159:
9066:
8755:
8722:
8627:
8509:
8443:
8418:
8018:
7800:
7676:
7621:
7476:
7441:
7406:
7336:
7293:
6478:
5520:
2633:
1374:
1056:
and this is the sine-Gordon equation, after scaling time and distance appropriately.
788:
768:
99:
7684:
6816:
Frenkel J, Kontorova T (1939). "On the theory of plastic deformation and twinning".
9122:
9014:
8997:
8750:
8659:
8453:
8363:
8081:
8023:
8008:
734:
138:
6565:{\displaystyle \partial _{t}u={\frac {1}{2}}\Delta u+c\sin(\beta u+\theta )+\xi ,}
2571:
7768:
7222:
McKean, H. P. (1981). "The sine-Gordon and sinh-Gordon equations on the circle".
6786:
5620:. More precisely, Liouville field theory is the Toda field theory for the finite
8948:
8111:
7890:
Faddeev, L. D. (1996). "How
Algebraic Bethe Ansatz works for integrable model".
7208:
6976:
6766:
6721:
6625:
6248:
201:
130:
6877:. North-Holland Personal Library. Vol. 15. North-Holland. pp. 34â45.
5947:
The quantum sine-Gordon equation should be modified so the exponentials become
2239:. The form of the 1-soliton solutions can be obtained through application of a
168:
This equation attracted a lot of attention in the 1970s due to the presence of
7876:
7613:
784:
489:
7753:
7515:
3411:
is the breather's frequency. If the old position of the standing breather is
9117:
8803:
8649:
8592:
8244:
6802:
5754:
7235:
5726:
the sine-Gordon model contains a parameter that can be identified with the
3503:
3485:
3191:
3172:
3158:
2971:
2958:
2555:
2541:
886:
17:
8342:
7938:
7075:
6856:
5731:
5222:
3198: – looks exotic, but essentially has a breather envelope.
2988:
2637:
176:. Among well-known integrable PDEs, the sine-Gordon equation is the only
6971:, vol. 10, Cham: Springer International Publishing, pp. 1â30,
8814:
7896:
7840:
7816:"Renormalization Group Theory of the Interfacial Roughening Transition"
7815:
7792:
7659:
7371:
7328:
7268:
3862:
are solutions to the same equation, that is, the sine-Gordon equation.
2620:
1829:
1817:
1816:
An interesting feature of the sine-Gordon equation is the existence of
169:
153:. The equation was rediscovered by Frenkel and Kontorova (
7026:
2084:
The 1-soliton solution for which we have chosen the positive root for
8542:
7420:
Faddeev, L. D.; Korepin, V. E. (1978). "Quantum theory of solitons".
6761:
1378:
5541:
Another similar equation comes from the EulerâLagrange equation for
313:
where partial derivatives are denoted by subscripts. Passing to the
200:
There are two equivalent forms of the sine-Gordon equation. In the (
7563:
7542:
2458:) twist of the elastic ribbon to be a kink with topological charge
7596:
2641:
2570:
885:
750:
5737:
Semi-classical quantization of the sine-Gordon model was done by
2074:{\displaystyle \varphi _{tt}-\varphi _{xx}+m^{2}\sin \varphi =0.}
8546:
7942:
6682:
again play a role in determining convergence of certain terms.
2623:
solutions can be obtained through continued application of the
2004:
and the slightly more general form of the equation is assumed:
7162:
Polyanin, Andrei D.; Zaitsev, Valentin F. (16 December 2011).
197:
This is the first derivation of the equation, by Bour (1862).
7058:
5903:
is not renormalized. Further, for a critical, non-zero value
7492:"Quantum sine-Gordon equation as the massive Thirring model"
5963:
5327:
1698:
1516:
1267:
779:. Another transformation of pseudospherical surfaces is the
609:{\displaystyle ds^{2}=du^{2}+2\cos \varphi \,du\,dv+dv^{2},}
397:{\displaystyle u={\frac {x+t}{2}},\quad v={\frac {x-t}{2}},}
303:{\displaystyle \varphi _{tt}-\varphi _{xx}+\sin \varphi =0,}
7057:
Georgiev D. D.; Papaioanou S. N.; Glazebrook J. F. (2004).
6698:
The sine-Gordon model arises as the continuum limit of the
5607:{\displaystyle \varphi _{xx}-\varphi _{tt}=2e^{2\varphi }.}
5298:{\displaystyle \varphi _{xx}-\varphi _{tt}=\sinh \varphi .}
1178:
The name "sine-Gordon equation" is a pun on the well-known
5477:{\displaystyle \varphi _{xx}+\varphi _{yy}=\sin \varphi ,}
1163:{\displaystyle T\varphi _{xx}(1+\varphi _{x}^{2})^{-3/2}}
5025:{\displaystyle \partial _{v}A_{u}-\partial _{u}A_{v}+=0}
2563:
soliton represents a propagating counterclockwise twist.
7704:. Springer International Publishing. pp. 155â175.
7166:(Second ed.). Boca Raton: CRC Press. p. 485.
5657:, while sin(h)-Gordon is the Toda field theory for the
1236:{\displaystyle \varphi _{tt}-\varphi _{xx}+\varphi =0.}
4686:
4520:
3802:
is an arbitrary parameter, is solvable for a function
7121:
Miroshnichenko A. E., Vasiliev A. A., Dmitriev S. V.
6634:
6610:
6578:
6490:
6423:
6387:
6347:
6306:
6261:
6217:
6175:
6114:
5959:
5909:
5889:
5862:
5835:
5808:
5775:
5666:
5629:
5552:
5493:
5429:
5324:
5250:
5200:
5173:
5082:
5040:
4940:
4909:
4667:
4501:
4474:
4445:
4413:
4381:
4341:
4300:
4161:
4141:
3996:
3976:
3940:
3920:
3894:
3874:
3848:
3828:
3808:
3615:
3563:
3540:
3444:
3417:
3397:
3370:
3248:
3218:
3013:
2807:
2657:
2581:
2504:
2464:
2339:
2252:
2221:
2189:
2160:
2134:
2114:
2090:
2013:
1949:
1841:
1508:
1390:
1263:
1191:
1098:
1065:
944:
924:
904:
853:
821:
801:
683:
629:
528:
498:
416:
338:
249:
214:
112:
84:
64:
44:
7164:
Handbook of Nonlinear Partial Differential Equations
7094:. Cambridge Texts in Applied Mathematics. New York:
6900:
Handbook of Nonlinear Partial Differential Equations
9152:
9131:
9095:
9044:
9037:
9006:
8985:
8978:
8938:
8906:
8865:
8835:
8791:
8743:
8737:
Integrable PDEs/Classical integrable field theories
8736:
8710:
8680:
8615:
8608:
8580:
8406:
8356:
8335:
8284:
8258:
8222:
8186:
8135:
8054:
8047:
7976:
492:on the surface. At every point on the surface, let
7005:Rubinstein, Julio (1970). "Sine-Gordon equation".
6902:. Chapman & Hall/CRC Press. pp. 470â492.
6674:
6616:
6596:
6564:
6445:
6409:
6369:
6333:
6292:
6239:
6188:
6149:
6097:
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5821:
5794:
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5068:
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4281:
4147:
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3834:
3814:
3787:
3595:
3546:
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3430:
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3383:
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3231:
3144:
2944:
2791:
2603:
2526:
2486:
2440:
2322:
2227:
2207:
2175:
2146:
2128:which takes the system from one constant solution
2120:
2096:
2073:
1993:
1929:
1800:
1484:
1362:
1235:
1162:
1084:
1048:
930:
910:
859:
839:
807:
755:Lie transform applied to pseudosphere to obtain a
714:
665:
608:
504:
447:
396:
302:
232:
118:
90:
70:
50:
7935:at NEQwiki, the nonlinear equations encyclopedia.
6898:Polyanin, Andrei D.; Valentin F. Zaitsev (2004).
6818:Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya
3784:
3774:
3749:
3689:
3664:
2549:soliton represents a propagating clockwise twist.
1994:{\displaystyle \gamma ^{2}={\frac {1}{1-v^{2}}},}
7922:at EqWorld: The World of Mathematical Equations.
7916:at EqWorld: The World of Mathematical Equations.
6675:{\displaystyle \beta ^{2}={\frac {n}{n+1}}8\pi }
6251:are needed to render the theory well-posed. The
6150:{\displaystyle V_{\beta }=:e^{i\beta \varphi }:}
3166:is an oscillating coupled kink-antikink soliton.
154:
8926:Six-dimensional holomorphic ChernâSimons theory
7141:. Oxford: Oxford University Press. p. 49.
6868:
6866:
6377:, the theory becomes ill-defined (Coleman
3438:, after the collision the new position will be
7249:
7247:
7245:
7224:Communications on Pure and Applied Mathematics
7052:
7050:
7048:
7046:
7044:
7042:
7040:
7038:
7036:
4331:The sine-Gordon equation is equivalent to the
3596:{\displaystyle \varphi _{uv}=\sin \varphi .\,}
2627:to the 1-soliton solution, as prescribed by a
8558:
7954:
6736:for vortices can therefore be derived from a
3954:related to the boost applied to the soliton.
2801:while the kink-antikink solution is given by
2636:. Since the colliding solitons recover their
1828:The sine-Gordon equation has the following 1-
58:dependent on two variables typically denoted
8:
7640:
7638:
6948:: CS1 maint: multiple names: authors list (
6293:{\displaystyle 4\pi <\beta ^{2}<8\pi }
2992:. There are known three types of breathers:
791:for solutions of the sine-Gordon equation.
715:{\displaystyle \varphi _{uv}=\sin \varphi .}
448:{\displaystyle \varphi _{uv}=\sin \varphi .}
7578:Hairer, Martin; Shen, Hao (February 2016).
5856:receives only a multiplicative correction,
5714:in addition to the solitons and breathers.
5694:{\displaystyle {\hat {\mathfrak {sl}}}_{2}}
5069:{\displaystyle \varphi _{uv}=\sin \varphi }
3007:The standing breather solution is given by
744:. There is a theorem, sometimes called the
512:be the angle between the asymptotic lines.
9153:Classical and quantum statistical lattices
9041:
8982:
8903:
8740:
8612:
8565:
8551:
8543:
8051:
7961:
7947:
7939:
6969:The sine-Gordon Model and its Applications
6923:Terng, C. L., & Uhlenbeck, K. (2000).
6791:Journal de l'Ăcole impĂ©riale polytechnique
6740:analysis of the sine-Gordon field theory.
5883:receives only an additive correction, and
5034:is equivalent to the sine-Gordon equation
3554:is a solution of the sine-Gordon equation
3125:
3095:
3071:
7895:
7839:
7658:
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2018:
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373:
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270:
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213:
111:
83:
63:
43:
7814:Ohta, T.; Kawasaki, K. (1 August 1978).
6837:Journal of the Physical Society of Japan
6787:"Theorie de la deformation des surfaces"
5412:Another closely related equation is the
3502:
3484:
3471:{\displaystyle x_{0}+\Delta _{\text{B}}}
3190:
3171:
3157:
2970:
2957:
2554:
2540:
188:Realizations of the sine-Gordon equation
6777:
6378:
2108:and represents a twist in the variable
139:surfaces of constant negative curvature
36:nonlinear partial differential equation
27:Nonlinear partial differential equation
8039:Two-dimensional conformal field theory
7773:Communications in Mathematical Physics
7584:Communications in Mathematical Physics
7117:
7115:
6941:
6169:For different values of the parameter
5929:{\displaystyle \beta ={\sqrt {4\pi }}}
2527:{\displaystyle \theta _{\text{AK}}=+1}
6732:, which is a model of magnetism. The
6712:The sine-Gordon model is in the same
6334:{\displaystyle {\frac {n}{n+1}}8\pi }
2487:{\displaystyle \theta _{\text{K}}=-1}
666:{\displaystyle L=N=0,M=\sin \varphi }
7:
8979:Exactly solvable quantum spin chains
8914:Four-dimensional ChernâSimons theory
8872:
7309:Theoretical and Mathematical Physics
6728:and anti-vortices in the continuous
6161:. A possible mass term is included.
5650:{\displaystyle {\mathfrak {sl}}_{2}}
2494:. The alternative counterclockwise (
2208:{\displaystyle \varphi \cong 2\pi n}
134:
8843:Anti-self-dual YangâMills equations
6702:which models crystal dislocations.
6370:{\displaystyle \beta ^{2}>8\pi }
6240:{\displaystyle \beta ^{2}<4\pi }
5675:
5672:
5636:
5633:
5507:is now a function of the variables
4934:. Then the zero-curvature equation
4364:{\displaystyle {\mathfrak {su}}(2)}
4347:
4344:
2651:The kink-kink solution is given by
2644:, such an interaction is called an
172:solutions, and is an example of an
8698:Superintegrable Hamiltonian system
7139:Solitons, instantons, and twistors
6517:
6492:
6007:
5994:
5936:, the theory is in fact dual to a
5795:{\displaystyle \alpha _{0},\beta }
5705:Infinite volume and on a half line
5129:
5103:
4965:
4942:
4101:
4071:
3459:
3250:
3232:{\displaystyle \Delta _{\text{B}}}
3220:
3002:traveling small-amplitude breather
2998:traveling large-amplitude breather
1737:
1631:
1425:
25:
9020:Quantum inverse scattering method
8866:Integrable Quantum Field theories
8534:Template:Quantum mechanics topics
7580:"The Dynamical Sine-Gordon Model"
7490:Coleman, Sidney (15 April 1975).
7090:Rogers, C.; W. K. Schief (2002).
5538:may be defined in a similar way.
3934:is the one-soliton solution with
3391:is the velocity of the kink, and
2154:to an adjacent constant solution
9045:Classical mechanics and geometry
8529:
8528:
6686:Supersymmetric sine-Gordon model
6597:{\displaystyle c,\beta ,\theta }
6446:{\displaystyle \beta ^{2}=8\pi }
6410:{\displaystyle \beta ^{2}=4\pi }
4461:{\displaystyle \mathbb {R} ^{2}}
4397:{\displaystyle \mathbb {R} ^{2}}
3907:{\displaystyle \varphi \equiv 0}
2498:) twist with topological charge
1246:The sine-Gordon equation is the
129:It was originally introduced by
8783:KadomtsevâPetviashvili equation
7820:Progress of Theoretical Physics
7767:Fröhlich, JĂŒrg (October 1976).
7352:Progress of Theoretical Physics
7124:Solitons and Soliton Collisions
7007:Journal of Mathematical Physics
6747:, in particular the XXZ model.
6604:are real-valued constants, and
6157:, where the semi-colons denote
3196:Small-amplitude moving breather
3177:Large-amplitude moving breather
840:{\displaystyle \varphi +2n\pi }
746:fundamental theorem of surfaces
366:
8768:Nonlinear Schrödinger equation
7256:Journal of High Energy Physics
6734:KosterlitzâThouless transition
6550:
6535:
6092:
6063:
5679:
5418:Euclidean sine-Gordon equation
5381:
5345:
5151:
5099:
5013:
4987:
4426:
4414:
4358:
4352:
4271:
4253:
4237:
4201:
4110:
4089:
4080:
4062:
3322:
3298:
3295:
3276:
3129:
3102:
3087:
3078:
3029:
3017:
2841:
2829:
2688:
2676:
2604:{\displaystyle 4\arctan e^{x}}
2176:{\displaystyle \varphi =2\pi }
1909:
1894:
1864:
1852:
1782:
1773:
1762:
1745:
1715:
1709:
1676:
1667:
1656:
1639:
1589:
1553:
1533:
1527:
1470:
1461:
1450:
1433:
1403:
1397:
1336:
1300:
1284:
1278:
1140:
1115:
1085:{\displaystyle T\varphi _{xx}}
787:in 1879, which corresponds to
227:
215:
1:
8939:Exactly solvable statistical
7286:10.1088/1126-6708/2007/03/047
5769:: there are three parameters
5616:A generalization is given by
5536:elliptic sinh-Gordon equation
5414:elliptic sine-Gordon equation
5157:{\displaystyle F_{\mu \nu }=}
4407:Explicitly, with coordinates
3958:Topological charge and energy
469: = â1, also called
8827:Inverse scattering transform
7677:10.1016/0370-2693(95)01072-X
7477:10.1016/0550-3213(78)90239-0
7442:10.1016/0370-1573(78)90058-3
7407:10.1016/0370-2693(85)90264-3
6465:Stochastic sine-Gordon model
6165:Regimes of renormalizability
4468:, the connection components
3384:{\displaystyle v_{\text{K}}}
1820:and multisoliton solutions.
407:the equation takes the form
137:) in the course of study of
8718:Quantum harmonic oscillator
8498:Quantum information science
6977:10.1007/978-3-319-06722-3_1
6475:dynamical sine-Gordon model
6381:). The boundary values are
6253:super-renormalizable regime
5876:{\displaystyle \gamma _{0}}
5849:{\displaystyle \alpha _{0}}
5822:{\displaystyle \gamma _{0}}
4923:{\displaystyle \sigma _{i}}
1495:it can be rewritten as the
9241:
8951:in one- and two-dimensions
7096:Cambridge University Press
6189:{\displaystyle \beta ^{2}}
4327:Zero-curvature formulation
3527:
2575:Static 1-soliton solution
2147:{\displaystyle \varphi =0}
874:
815:is a solution, then so is
9062:Ferdinand Georg Frobenius
8667:Garnier integrable system
8523:
7877:10.1103/RevModPhys.51.659
7857:Reviews of Modern Physics
7614:10.1007/s00220-015-2525-3
7209:10.1393/ncb/i2005-10164-6
7137:Dunajski, Maciej (2010).
5718:Quantum sine-Gordon model
1499:plus higher-order terms:
728:Hilbert embedding theorem
145:for surfaces of constant
8993:Quantum Heisenberg model
8836:ASDYM as a master theory
8688:LiouvilleâArnold theorem
7754:10.1103/PhysRevD.14.2826
7516:10.1103/PhysRevD.11.2088
6745:quantum Heisenberg model
6707:long Josephson junctions
5659:affine KacâMoody algebra
5500:{\displaystyle \varphi }
4488:{\displaystyle A_{\mu }}
4148:{\displaystyle \varphi }
3983:{\displaystyle \varphi }
3888:is the trivial solution
3881:{\displaystyle \varphi }
3835:{\displaystyle \varphi }
3547:{\displaystyle \varphi }
2121:{\displaystyle \varphi }
931:{\displaystyle \varphi }
808:{\displaystyle \varphi }
777:BĂ€cklund transformations
775:led to the discovery of
730:. In the simplest case,
505:{\displaystyle \varphi }
471:pseudospherical surfaces
119:{\displaystyle \varphi }
51:{\displaystyle \varphi }
9215:Exactly solvable models
9139:Alexander Zamolodchikov
8799:BĂ€cklund transformation
8728:PöschlâTeller potential
8600:Liouville integrability
8588:Frobenius integrability
8581:Geometric integrability
8194:2D free massless scalar
8087:Quantum electrodynamics
8014:QFT in curved spacetime
6700:FrenkelâKontorova model
5751:Alexander Zamolodchikov
5310:EulerâLagrange equation
3524:BĂ€cklund transformation
3404:{\displaystyle \omega }
2228:{\displaystyle \gamma }
2097:{\displaystyle \gamma }
1497:KleinâGordon Lagrangian
1248:EulerâLagrange equation
973:mass times acceleration
877:FrenkelâKontorova model
871:FrenkelâKontorova model
621:second fundamental form
484: = constant,
163:FrenkelâKontorova model
8898:Principal chiral model
8848:Twistor correspondence
8693:Action-angle variables
8609:In classical mechanics
8515:Quantum thermodynamics
8439:On shell and off shell
8434:Loop quantum cosmology
8276:N = 4 super YangâMills
8235:N = 1 super YangâMills
8102:Scalar electrodynamics
8092:Quantum chromodynamics
7994:Conformal field theory
7970:Quantum field theories
7236:10.1002/cpa.3160340204
6925:"Geometry of solitons"
6873:Rajaraman, R. (1989).
6676:
6618:
6598:
6566:
6447:
6411:
6371:
6335:
6294:
6241:
6190:
6151:
6099:
5930:
5897:
5896:{\displaystyle \beta }
5877:
5850:
5823:
5796:
5695:
5651:
5608:
5543:Liouville field theory
5501:
5478:
5403:
5299:
5215:
5188:
5158:
5070:
5026:
4924:
4897:
4656:
4489:
4462:
4433:
4398:
4365:
4314:
4283:
4149:
4125:
3984:
3948:
3928:
3908:
3882:
3856:
3836:
3816:
3789:
3597:
3548:
3516:
3498:
3472:
3432:
3405:
3385:
3355:
3233:
3199:
3180:
3167:
3146:
2979:
2966:
2946:
2793:
2611:
2605:
2564:
2550:
2528:
2488:
2442:
2324:
2229:
2209:
2177:
2148:
2122:
2098:
2075:
1995:
1931:
1802:
1741:
1635:
1486:
1429:
1364:
1237:
1164:
1092:, but more accurately
1086:
1050:
932:
912:
895:
861:
841:
809:
763:New solutions from old
759:
716:
675:GaussâCodazzi equation
667:
610:
517:first fundamental form
506:
449:
398:
327:asymptotic coordinates
315:light-cone coordinates
304:
240:, the equation reads:
234:
206:space-time coordinates
143:GaussâCodazzi equation
120:
92:
72:
52:
9205:Differential geometry
9077:Joseph-Louis Lagrange
8628:Central force systems
8488:Quantum hydrodynamics
8483:Quantum hadrodynamics
8107:Scalar chromodynamics
6785:Bour, Edmond (1862).
6738:renormalization group
6694:Physical applications
6677:
6619:
6599:
6567:
6448:
6412:
6372:
6336:
6295:
6242:
6191:
6152:
6100:
5931:
5898:
5878:
5851:
5824:
5797:
5712:boundary bound states
5696:
5652:
5609:
5517:analytic continuation
5502:
5479:
5404:
5300:
5216:
5214:{\displaystyle A_{v}}
5189:
5187:{\displaystyle A_{u}}
5167:The pair of matrices
5159:
5071:
5027:
4925:
4898:
4657:
4490:
4463:
4434:
4432:{\displaystyle (u,v)}
4404:being equal to zero.
4399:
4366:
4315:
4284:
4150:
4126:
3985:
3949:
3929:
3927:{\displaystyle \psi }
3909:
3883:
3857:
3855:{\displaystyle \psi }
3837:
3817:
3815:{\displaystyle \psi }
3790:
3598:
3549:
3506:
3488:
3473:
3433:
3431:{\displaystyle x_{0}}
3406:
3386:
3356:
3234:
3194:
3175:
3161:
3147:
2974:
2961:
2947:
2794:
2606:
2574:
2558:
2544:
2534:will be an antikink.
2529:
2489:
2443:
2325:
2230:
2210:
2178:
2149:
2123:
2099:
2076:
1996:
1932:
1803:
1721:
1615:
1487:
1409:
1365:
1238:
1180:KleinâGordon equation
1165:
1087:
1051:
933:
913:
889:
862:
842:
810:
754:
742:rigid transformations
717:
668:
611:
507:
450:
399:
305:
235:
233:{\displaystyle (x,t)}
193:Differential geometry
121:
93:
73:
53:
9225:Mathematical physics
9220:Equations of physics
9144:Alexei Zamolodchikov
9113:Martin David Kruskal
9087:Siméon Denis Poisson
9025:YangâBaxter equation
8920:Affine Gaudin models
8763:Sine-Gordon equation
8711:In quantum mechanics
8459:Quantum fluctuations
8429:Loop quantum gravity
7999:Lattice field theory
7926:sine-Gordon equation
7920:Sinh-Gordon Equation
7914:sine-Gordon equation
7076:10.14748/bmr.v15.103
6857:10.1143/JPSJ.33.1459
6632:
6617:{\displaystyle \xi }
6608:
6576:
6488:
6477:has been studied by
6421:
6385:
6345:
6304:
6259:
6215:
6173:
6112:
5957:
5907:
5887:
5860:
5833:
5806:
5773:
5745:. The exact quantum
5724:quantum field theory
5664:
5627:
5550:
5491:
5427:
5322:
5248:
5237:sinh-Gordon equation
5221:are also known as a
5198:
5171:
5080:
5038:
4938:
4907:
4665:
4499:
4472:
4443:
4411:
4379:
4339:
4298:
4159:
4139:
3994:
3974:
3938:
3918:
3892:
3872:
3846:
3826:
3806:
3613:
3561:
3538:
3442:
3415:
3395:
3368:
3246:
3216:
3011:
2805:
2655:
2579:
2502:
2462:
2337:
2250:
2219:
2187:
2158:
2132:
2112:
2088:
2011:
1947:
1839:
1506:
1388:
1261:
1189:
1096:
1063:
942:
922:
902:
851:
819:
799:
681:
627:
526:
496:
414:
336:
247:
212:
159:crystal dislocations
157:) in their study of
110:
82:
62:
42:
32:sine-Gordon equation
9103:Clifford S. Gardner
8873:Quantum Sine-Gordon
8820:Topological soliton
8810:integrals of motion
8623:Harmonic oscillator
8493:Quantum information
8097:Quartic interaction
7869:1979RvMP...51..659K
7832:1978PThPh..60..365O
7785:1976CMaPh..47..233F
7746:1976PhRvD..14.2826J
7669:1995PhLB..359..118I
7606:2016CMaPh.341..933H
7508:1975PhRvD..11.2088C
7469:1978NuPhB.133..525Z
7434:1978PhR....42....1F
7399:1985PhLB..159..345B
7364:1981PThPh..66..101T
7321:1979TMP....41..953K
7278:2007JHEP...03..047B
7201:2006NCimB.121..115X
7019:1970JMP....11..258R
6849:1972JPSJ...33.1459H
6046:
5761:, as discovered by
5380:
5362:
4313:{\displaystyle N=0}
4236:
4218:
3321:
3207:3-soliton solutions
2615:2-soliton solutions
2352:
2265:
1824:1-soliton solutions
1588:
1570:
1381:in the Lagrangian,
1335:
1317:
1250:of the field whose
1138:
151:3-dimensional space
9072:Sofia Kovalevskaya
8970:Chiral Potts model
8965:Hard hexagon model
8960:Eight-vertex model
8574:Integrable systems
8379:NambuâJona-Lasinio
8307:Higher dimensional
8214:WessâZuminoâWitten
8004:Noncommutative QFT
7931:2012-03-16 at the
7841:10.1143/PTP.60.365
7793:10.1007/BF01609843
7372:10.1143/ptp.66.101
7329:10.1007/bf01028501
7189:Il Nuovo Cimento B
7063:Biomedical Reviews
6932:Notices of the AMS
6730:classical XY model
6714:universality class
6672:
6614:
6594:
6562:
6443:
6407:
6367:
6331:
6290:
6237:
6186:
6147:
6095:
6032:
5942:Dirac field theory
5926:
5893:
5873:
5846:
5819:
5792:
5749:was discovered by
5691:
5647:
5604:
5497:
5474:
5399:
5366:
5348:
5295:
5211:
5184:
5154:
5066:
5022:
4920:
4893:
4801:
4652:
4588:
4485:
4458:
4429:
4394:
4361:
4310:
4279:
4222:
4204:
4145:
4121:
3980:
3964:topological charge
3944:
3924:
3904:
3878:
3852:
3832:
3812:
3785:
3781:
3593:
3544:
3530:BĂ€cklund transform
3517:
3499:
3468:
3428:
3401:
3381:
3351:
3307:
3229:
3200:
3181:
3168:
3142:
2980:
2967:
2942:
2789:
2625:BĂ€cklund transform
2612:
2601:
2565:
2551:
2524:
2484:
2438:
2340:
2320:
2253:
2241:BĂ€cklund transform
2225:
2205:
2173:
2144:
2118:
2094:
2071:
1991:
1927:
1798:
1796:
1574:
1556:
1482:
1360:
1321:
1303:
1252:Lagrangian density
1233:
1160:
1124:
1082:
1046:
1045:
1038:
1011:
1004:
976:
969:
928:
908:
896:
882:A mechanical model
857:
837:
805:
760:
712:
663:
606:
519:of the surface is
502:
464:Gaussian curvature
445:
394:
300:
230:
182:Lorentz invariance
180:system due to its
147:Gaussian curvature
116:
88:
68:
48:
34:is a second-order
9187:
9186:
9183:
9182:
9033:
9032:
8934:
8933:
8893:Toda field theory
8883:Quantum Liouville
8861:
8860:
8815:Soliton solutions
8773:GrossâNeveu model
8706:
8705:
8540:
8539:
8402:
8401:
7740:(10): 2826â2829.
7734:Physical Review D
7711:978-3-319-06722-3
7647:Physics Letters B
7496:Physical Review D
7457:Nuclear Physics B
7387:Physics Letters B
7173:978-1-4200-8723-9
7148:978-0-19-857063-9
7105:978-0-521-01288-1
7027:10.1063/1.1665057
6986:978-3-319-06721-6
6909:978-1-58488-355-5
6884:978-0-444-87047-6
6664:
6515:
6323:
6198:renormalizability
6030:
5991:
5924:
5829:. Coleman showed
5747:scattering matrix
5682:
5622:KacâMoody algebra
5618:Toda field theory
5343:
5229:Related equations
4866:
4826:
4788:
4762:
4734:
4705:
4605:
4563:
4539:
4199:
4052:
4016:
3947:{\displaystyle a}
3770:
3739:
3685:
3521:
3520:
3513:standing breather
3495:standing breather
3465:
3378:
3346:
3345:
3325:
3314:
3256:
3226:
3204:
3203:
3185:
3184:
3164:standing breather
3133:
3123:
3069:
2994:standing breather
2984:
2983:
2936:
2933:
2932:
2895:
2894:
2783:
2780:
2779:
2742:
2741:
2646:elastic collision
2569:
2568:
2512:
2472:
2414:
2409:
2380:
2315:
1986:
1849:
1812:Soliton solutions
1789:
1706:
1683:
1610:
1551:
1524:
1477:
1377:expansion of the
1298:
1275:
1043:
1017:
1015:
1009:
982:
980:
974:
947:
945:
911:{\displaystyle x}
860:{\displaystyle n}
478:asymptotic curves
389:
361:
91:{\displaystyle t}
71:{\displaystyle x}
16:(Redirected from
9232:
9082:Joseph Liouville
9042:
8983:
8955:Square ice model
8904:
8808:Infinitely many
8741:
8638:Two body problem
8613:
8567:
8560:
8553:
8544:
8532:
8531:
8449:Quantum dynamics
8122:YangâMillsâHiggs
8077:Non-linear sigma
8067:EulerâHeisenberg
8052:
7963:
7956:
7949:
7940:
7902:
7901:
7899:
7887:
7881:
7880:
7852:
7846:
7845:
7843:
7811:
7805:
7804:
7764:
7758:
7757:
7729:
7723:
7722:
7720:
7718:
7695:
7689:
7688:
7662:
7642:
7633:
7632:
7630:
7628:
7599:
7575:
7569:
7568:
7566:
7554:
7548:
7547:
7545:
7533:
7527:
7526:
7524:
7522:
7502:(8): 2088â2097.
7487:
7481:
7480:
7452:
7446:
7445:
7417:
7411:
7410:
7382:
7376:
7375:
7347:
7341:
7340:
7304:
7298:
7297:
7271:
7251:
7240:
7239:
7219:
7213:
7212:
7184:
7178:
7177:
7159:
7153:
7152:
7134:
7128:
7119:
7110:
7109:
7087:
7081:
7080:
7078:
7054:
7031:
7030:
7002:
6996:
6995:
6994:
6993:
6960:
6954:
6953:
6947:
6939:
6929:
6920:
6914:
6913:
6895:
6889:
6888:
6870:
6861:
6860:
6843:(5): 1459â1463.
6832:
6826:
6825:
6813:
6807:
6806:
6782:
6757:Josephson effect
6718:effective action
6681:
6679:
6678:
6673:
6665:
6663:
6649:
6644:
6643:
6623:
6621:
6620:
6615:
6603:
6601:
6600:
6595:
6571:
6569:
6568:
6563:
6516:
6508:
6500:
6499:
6484:The equation is
6452:
6450:
6449:
6444:
6433:
6432:
6416:
6414:
6413:
6408:
6397:
6396:
6376:
6374:
6373:
6368:
6357:
6356:
6340:
6338:
6337:
6332:
6324:
6322:
6308:
6299:
6297:
6296:
6291:
6280:
6279:
6246:
6244:
6243:
6238:
6227:
6226:
6195:
6193:
6192:
6187:
6185:
6184:
6156:
6154:
6153:
6148:
6143:
6142:
6124:
6123:
6104:
6102:
6101:
6096:
6091:
6090:
6075:
6074:
6056:
6055:
6045:
6040:
6031:
6023:
6015:
6014:
6002:
6001:
5992:
5984:
5979:
5978:
5967:
5966:
5949:vertex operators
5935:
5933:
5932:
5927:
5925:
5917:
5902:
5900:
5899:
5894:
5882:
5880:
5879:
5874:
5872:
5871:
5855:
5853:
5852:
5847:
5845:
5844:
5828:
5826:
5825:
5820:
5818:
5817:
5801:
5799:
5798:
5793:
5785:
5784:
5753:. This model is
5743:Vladimir Korepin
5700:
5698:
5697:
5692:
5690:
5689:
5684:
5683:
5678:
5670:
5656:
5654:
5653:
5648:
5646:
5645:
5640:
5639:
5613:
5611:
5610:
5605:
5600:
5599:
5581:
5580:
5565:
5564:
5506:
5504:
5503:
5498:
5483:
5481:
5480:
5475:
5458:
5457:
5442:
5441:
5408:
5406:
5405:
5400:
5379:
5374:
5361:
5356:
5344:
5336:
5331:
5330:
5304:
5302:
5301:
5296:
5279:
5278:
5263:
5262:
5239:
5238:
5220:
5218:
5217:
5212:
5210:
5209:
5193:
5191:
5190:
5185:
5183:
5182:
5163:
5161:
5160:
5155:
5150:
5149:
5137:
5136:
5124:
5123:
5111:
5110:
5095:
5094:
5075:
5073:
5072:
5067:
5053:
5052:
5031:
5029:
5028:
5023:
5012:
5011:
4999:
4998:
4983:
4982:
4973:
4972:
4960:
4959:
4950:
4949:
4929:
4927:
4926:
4921:
4919:
4918:
4902:
4900:
4899:
4894:
4889:
4888:
4867:
4865:
4854:
4849:
4848:
4827:
4825:
4814:
4806:
4805:
4789:
4787:
4776:
4763:
4761:
4750:
4735:
4733:
4722:
4706:
4704:
4693:
4677:
4676:
4661:
4659:
4658:
4653:
4648:
4647:
4629:
4628:
4616:
4615:
4606:
4598:
4593:
4592:
4574:
4573:
4564:
4556:
4550:
4549:
4540:
4532:
4511:
4510:
4494:
4492:
4491:
4486:
4484:
4483:
4467:
4465:
4464:
4459:
4457:
4456:
4451:
4438:
4436:
4435:
4430:
4403:
4401:
4400:
4395:
4393:
4392:
4387:
4370:
4368:
4367:
4362:
4351:
4350:
4335:of a particular
4319:
4317:
4316:
4311:
4288:
4286:
4285:
4280:
4278:
4274:
4252:
4251:
4235:
4230:
4217:
4212:
4200:
4192:
4179:
4178:
4177:
4154:
4152:
4151:
4146:
4130:
4128:
4127:
4122:
4117:
4113:
4053:
4051:
4040:
4029:
4028:
4027:
4017:
4015:
4004:
3989:
3987:
3986:
3981:
3953:
3951:
3950:
3945:
3933:
3931:
3930:
3925:
3913:
3911:
3910:
3905:
3887:
3885:
3884:
3879:
3868:For example, if
3861:
3859:
3858:
3853:
3841:
3839:
3838:
3833:
3821:
3819:
3818:
3813:
3794:
3792:
3791:
3786:
3782:
3778:
3777:
3771:
3766:
3755:
3753:
3752:
3740:
3732:
3727:
3726:
3707:
3706:
3693:
3692:
3686:
3681:
3670:
3668:
3667:
3646:
3645:
3629:
3628:
3606:Then the system
3602:
3600:
3599:
3594:
3576:
3575:
3553:
3551:
3550:
3545:
3481:
3480:
3477:
3475:
3474:
3469:
3467:
3466:
3463:
3454:
3453:
3437:
3435:
3434:
3429:
3427:
3426:
3410:
3408:
3407:
3402:
3390:
3388:
3387:
3382:
3380:
3379:
3376:
3360:
3358:
3357:
3352:
3347:
3344:
3343:
3328:
3327:
3326:
3320:
3315:
3312:
3294:
3293:
3275:
3263:
3258:
3257:
3254:
3238:
3236:
3235:
3230:
3228:
3227:
3224:
3187:
3186:
3154:
3153:
3151:
3149:
3148:
3143:
3138:
3134:
3132:
3124:
3122:
3121:
3106:
3090:
3070:
3068:
3067:
3052:
3049:
2954:
2953:
2951:
2949:
2948:
2943:
2941:
2937:
2935:
2934:
2931:
2930:
2915:
2914:
2906:
2897:
2896:
2893:
2892:
2877:
2873:
2861:
2828:
2827:
2820:
2798:
2796:
2795:
2790:
2788:
2784:
2782:
2781:
2778:
2777:
2762:
2761:
2753:
2744:
2743:
2740:
2739:
2724:
2720:
2708:
2675:
2674:
2670:
2610:
2608:
2607:
2602:
2600:
2599:
2537:
2536:
2533:
2531:
2530:
2525:
2514:
2513:
2510:
2493:
2491:
2490:
2485:
2474:
2473:
2470:
2447:
2445:
2444:
2439:
2431:
2430:
2415:
2413: with
2412:
2410:
2405:
2398:
2389:
2381:
2373:
2368:
2367:
2348:
2329:
2327:
2326:
2321:
2316:
2311:
2304:
2295:
2278:
2277:
2261:
2234:
2232:
2231:
2226:
2214:
2212:
2211:
2206:
2182:
2180:
2179:
2174:
2153:
2151:
2150:
2145:
2127:
2125:
2124:
2119:
2103:
2101:
2100:
2095:
2080:
2078:
2077:
2072:
2055:
2054:
2042:
2041:
2026:
2025:
2000:
1998:
1997:
1992:
1987:
1985:
1984:
1983:
1964:
1959:
1958:
1936:
1934:
1933:
1928:
1923:
1919:
1918:
1851:
1850:
1847:
1807:
1805:
1804:
1799:
1797:
1790:
1788:
1771:
1770:
1769:
1760:
1759:
1743:
1740:
1735:
1708:
1707:
1704:
1702:
1701:
1688:
1684:
1682:
1665:
1664:
1663:
1654:
1653:
1637:
1634:
1629:
1611:
1606:
1605:
1596:
1587:
1582:
1569:
1564:
1552:
1544:
1526:
1525:
1522:
1520:
1519:
1491:
1489:
1488:
1483:
1478:
1476:
1459:
1458:
1457:
1448:
1447:
1431:
1428:
1423:
1369:
1367:
1366:
1361:
1334:
1329:
1316:
1311:
1299:
1291:
1277:
1276:
1273:
1271:
1270:
1242:
1240:
1239:
1234:
1220:
1219:
1204:
1203:
1169:
1167:
1166:
1161:
1159:
1158:
1154:
1137:
1132:
1114:
1113:
1091:
1089:
1088:
1083:
1081:
1080:
1055:
1053:
1052:
1047:
1044:
1041:
1039:
1034:
1010:
1007:
1005:
1000:
999:
998:
975:
972:
970:
965:
964:
963:
937:
935:
934:
929:
917:
915:
914:
909:
866:
864:
863:
858:
846:
844:
843:
838:
814:
812:
811:
806:
721:
719:
718:
713:
696:
695:
672:
670:
669:
664:
615:
613:
612:
607:
602:
601:
557:
556:
541:
540:
511:
509:
508:
503:
454:
452:
451:
446:
429:
428:
403:
401:
400:
395:
390:
385:
374:
362:
357:
346:
309:
307:
306:
301:
278:
277:
262:
261:
239:
237:
236:
231:
125:
123:
122:
117:
98:, involving the
97:
95:
94:
89:
77:
75:
74:
69:
57:
55:
54:
49:
21:
9240:
9239:
9235:
9234:
9233:
9231:
9230:
9229:
9190:
9189:
9188:
9179:
9170:Elliott H. Lieb
9148:
9127:
9091:
9052:Vladimir Arnold
9029:
9002:
8974:
8930:
8907:Master theories
8902:
8857:
8853:Ward conjecture
8831:
8787:
8732:
8702:
8676:
8645:Integrable tops
8604:
8576:
8571:
8541:
8536:
8519:
8471:Quantum gravity
8398:
8357:Particle theory
8352:
8331:
8280:
8254:
8218:
8182:
8136:Low dimensional
8131:
8072:GinzburgâLandau
8043:
8034:Topological QFT
7972:
7967:
7933:Wayback Machine
7910:
7905:
7889:
7888:
7884:
7854:
7853:
7849:
7813:
7812:
7808:
7766:
7765:
7761:
7731:
7730:
7726:
7716:
7714:
7712:
7697:
7696:
7692:
7644:
7643:
7636:
7626:
7624:
7577:
7576:
7572:
7556:
7555:
7551:
7535:
7534:
7530:
7520:
7518:
7489:
7488:
7484:
7454:
7453:
7449:
7422:Physics Reports
7419:
7418:
7414:
7384:
7383:
7379:
7349:
7348:
7344:
7306:
7305:
7301:
7253:
7252:
7243:
7221:
7220:
7216:
7186:
7185:
7181:
7174:
7161:
7160:
7156:
7149:
7136:
7135:
7131:
7120:
7113:
7106:
7089:
7088:
7084:
7056:
7055:
7034:
7004:
7003:
6999:
6991:
6989:
6987:
6962:
6961:
6957:
6940:
6927:
6922:
6921:
6917:
6910:
6897:
6896:
6892:
6885:
6872:
6871:
6864:
6834:
6833:
6829:
6815:
6814:
6810:
6784:
6783:
6779:
6775:
6753:
6696:
6688:
6653:
6635:
6630:
6629:
6606:
6605:
6574:
6573:
6491:
6486:
6485:
6467:
6458:
6424:
6419:
6418:
6388:
6383:
6382:
6348:
6343:
6342:
6312:
6302:
6301:
6271:
6257:
6256:
6218:
6213:
6212:
6176:
6171:
6170:
6167:
6159:normal ordering
6128:
6115:
6110:
6109:
6079:
6066:
6047:
6006:
5993:
5960:
5955:
5954:
5905:
5904:
5885:
5884:
5863:
5858:
5857:
5836:
5831:
5830:
5809:
5804:
5803:
5776:
5771:
5770:
5767:renormalization
5728:Planck constant
5720:
5707:
5667:
5662:
5661:
5630:
5625:
5624:
5588:
5569:
5553:
5548:
5547:
5489:
5488:
5446:
5430:
5425:
5424:
5320:
5319:
5267:
5251:
5246:
5245:
5236:
5235:
5231:
5201:
5196:
5195:
5174:
5169:
5168:
5141:
5128:
5115:
5102:
5083:
5078:
5077:
5041:
5036:
5035:
5003:
4990:
4974:
4964:
4951:
4941:
4936:
4935:
4910:
4905:
4904:
4880:
4858:
4840:
4818:
4800:
4799:
4780:
4773:
4754:
4746:
4745:
4726:
4716:
4697:
4682:
4668:
4663:
4662:
4639:
4620:
4607:
4587:
4586:
4575:
4565:
4552:
4551:
4541:
4529:
4516:
4502:
4497:
4496:
4475:
4470:
4469:
4446:
4441:
4440:
4409:
4408:
4382:
4377:
4376:
4337:
4336:
4329:
4323:
4296:
4295:
4290:interactions.
4243:
4190:
4186:
4168:
4157:
4156:
4137:
4136:
4058:
4054:
4044:
4018:
4008:
3992:
3991:
3972:
3971:
3960:
3936:
3935:
3916:
3915:
3890:
3889:
3870:
3869:
3844:
3843:
3824:
3823:
3804:
3803:
3780:
3779:
3756:
3718:
3708:
3698:
3695:
3694:
3671:
3637:
3630:
3620:
3611:
3610:
3564:
3559:
3558:
3536:
3535:
3532:
3526:
3509:moving antikink
3458:
3445:
3440:
3439:
3418:
3413:
3412:
3393:
3392:
3371:
3366:
3365:
3335:
3285:
3264:
3249:
3244:
3243:
3219:
3214:
3213:
3209:
3113:
3091:
3059:
3050:
3044:
3009:
3008:
2922:
2907:
2898:
2884:
2862:
2856:
2808:
2803:
2802:
2769:
2754:
2745:
2731:
2709:
2703:
2658:
2653:
2652:
2629:Bianchi lattice
2617:
2591:
2577:
2576:
2505:
2500:
2499:
2465:
2460:
2459:
2422:
2391:
2390:
2359:
2335:
2334:
2297:
2296:
2269:
2248:
2247:
2217:
2216:
2185:
2184:
2156:
2155:
2130:
2129:
2110:
2109:
2086:
2085:
2046:
2030:
2014:
2009:
2008:
1975:
1968:
1950:
1945:
1944:
1883:
1879:
1842:
1837:
1836:
1826:
1814:
1795:
1794:
1772:
1761:
1751:
1744:
1695:
1686:
1685:
1666:
1655:
1645:
1638:
1597:
1536:
1513:
1504:
1503:
1460:
1449:
1439:
1432:
1386:
1385:
1264:
1259:
1258:
1208:
1192:
1187:
1186:
1176:
1139:
1102:
1094:
1093:
1069:
1061:
1060:
1018:
987:
983:
952:
948:
940:
939:
920:
919:
900:
899:
884:
879:
873:
849:
848:
817:
816:
797:
796:
765:
684:
679:
678:
625:
624:
593:
548:
532:
524:
523:
494:
493:
417:
412:
411:
375:
347:
334:
333:
266:
250:
245:
244:
210:
209:
195:
190:
131:Edmond Bour
108:
107:
80:
79:
60:
59:
40:
39:
38:for a function
28:
23:
22:
15:
12:
11:
5:
9238:
9236:
9228:
9227:
9222:
9217:
9212:
9207:
9202:
9192:
9191:
9185:
9184:
9181:
9180:
9178:
9177:
9175:Yang Chen-Ning
9172:
9167:
9165:Ludvig Faddeev
9162:
9156:
9154:
9150:
9149:
9147:
9146:
9141:
9135:
9133:
9129:
9128:
9126:
9125:
9120:
9115:
9110:
9108:John M. Greene
9105:
9099:
9097:
9093:
9092:
9090:
9089:
9084:
9079:
9074:
9069:
9064:
9059:
9057:Leonhard Euler
9054:
9048:
9046:
9039:
9035:
9034:
9031:
9030:
9028:
9027:
9022:
9017:
9010:
9008:
9004:
9003:
9001:
9000:
8995:
8989:
8987:
8980:
8976:
8975:
8973:
8972:
8967:
8962:
8957:
8952:
8945:
8943:
8941:lattice models
8936:
8935:
8932:
8931:
8929:
8928:
8923:
8917:
8910:
8908:
8901:
8900:
8895:
8890:
8888:Thirring model
8885:
8880:
8875:
8869:
8867:
8863:
8862:
8859:
8858:
8856:
8855:
8850:
8845:
8839:
8837:
8833:
8832:
8830:
8829:
8824:
8823:
8822:
8812:
8806:
8801:
8795:
8793:
8789:
8788:
8786:
8785:
8780:
8778:Thirring model
8775:
8770:
8765:
8760:
8759:
8758:
8747:
8745:
8738:
8734:
8733:
8731:
8730:
8725:
8720:
8714:
8712:
8708:
8707:
8704:
8703:
8701:
8700:
8695:
8690:
8684:
8682:
8678:
8677:
8675:
8674:
8672:Hitchin system
8669:
8664:
8663:
8662:
8657:
8652:
8642:
8641:
8640:
8635:
8625:
8619:
8617:
8610:
8606:
8605:
8603:
8602:
8597:
8596:
8595:
8584:
8582:
8578:
8577:
8572:
8570:
8569:
8562:
8555:
8547:
8538:
8537:
8524:
8521:
8520:
8518:
8517:
8512:
8507:
8506:
8505:
8495:
8490:
8485:
8480:
8479:
8478:
8468:
8467:
8466:
8456:
8451:
8446:
8441:
8436:
8431:
8426:
8421:
8416:
8414:Casimir effect
8410:
8408:
8404:
8403:
8400:
8399:
8397:
8396:
8391:
8389:Standard Model
8386:
8381:
8376:
8371:
8366:
8360:
8358:
8354:
8353:
8351:
8350:
8345:
8339:
8337:
8333:
8332:
8330:
8329:
8324:
8319:
8314:
8309:
8304:
8299:
8294:
8288:
8286:
8282:
8281:
8279:
8278:
8273:
8268:
8262:
8260:
8259:Superconformal
8256:
8255:
8253:
8252:
8247:
8242:
8240:SeibergâWitten
8237:
8232:
8226:
8224:
8223:Supersymmetric
8220:
8219:
8217:
8216:
8211:
8206:
8201:
8196:
8190:
8188:
8184:
8183:
8181:
8180:
8175:
8170:
8165:
8160:
8155:
8150:
8145:
8139:
8137:
8133:
8132:
8130:
8129:
8124:
8119:
8114:
8109:
8104:
8099:
8094:
8089:
8084:
8079:
8074:
8069:
8064:
8058:
8056:
8049:
8045:
8044:
8042:
8041:
8036:
8031:
8026:
8021:
8016:
8011:
8006:
8001:
7996:
7991:
7986:
7980:
7978:
7974:
7973:
7968:
7966:
7965:
7958:
7951:
7943:
7937:
7936:
7923:
7917:
7909:
7908:External links
7906:
7904:
7903:
7897:hep-th/9605187
7882:
7863:(4): 659â713.
7847:
7826:(2): 365â379.
7806:
7779:(3): 233â268.
7759:
7724:
7710:
7690:
7660:hep-th/9506157
7653:(1): 118â124.
7634:
7590:(3): 933â989.
7570:
7549:
7528:
7482:
7463:(3): 525â535.
7447:
7412:
7393:(4): 345â347.
7377:
7358:(1): 101â117.
7342:
7315:(2): 953â967.
7299:
7269:hep-th/0203139
7241:
7230:(2): 197â257.
7214:
7195:(2): 115â121.
7179:
7172:
7154:
7147:
7129:
7111:
7104:
7082:
7032:
7013:(1): 258â266.
6997:
6985:
6955:
6915:
6908:
6890:
6883:
6862:
6827:
6808:
6776:
6774:
6771:
6770:
6769:
6764:
6759:
6752:
6749:
6695:
6692:
6687:
6684:
6671:
6668:
6662:
6659:
6656:
6652:
6647:
6642:
6638:
6624:is space-time
6613:
6593:
6590:
6587:
6584:
6581:
6561:
6558:
6555:
6552:
6549:
6546:
6543:
6540:
6537:
6534:
6531:
6528:
6525:
6522:
6519:
6514:
6511:
6506:
6503:
6498:
6494:
6466:
6463:
6456:
6442:
6439:
6436:
6431:
6427:
6406:
6403:
6400:
6395:
6391:
6366:
6363:
6360:
6355:
6351:
6330:
6327:
6321:
6318:
6315:
6311:
6289:
6286:
6283:
6278:
6274:
6270:
6267:
6264:
6236:
6233:
6230:
6225:
6221:
6183:
6179:
6166:
6163:
6146:
6141:
6138:
6135:
6131:
6127:
6122:
6118:
6106:
6105:
6094:
6089:
6086:
6082:
6078:
6073:
6069:
6065:
6062:
6059:
6054:
6050:
6044:
6039:
6035:
6029:
6026:
6021:
6018:
6013:
6009:
6005:
6000:
5996:
5990:
5987:
5982:
5977:
5974:
5971:
5965:
5923:
5920:
5915:
5912:
5892:
5870:
5866:
5843:
5839:
5816:
5812:
5791:
5788:
5783:
5779:
5759:Thirring model
5739:Ludwig Faddeev
5719:
5716:
5706:
5703:
5688:
5681:
5677:
5674:
5644:
5638:
5635:
5603:
5598:
5595:
5591:
5587:
5584:
5579:
5576:
5572:
5568:
5563:
5560:
5556:
5527: = i
5496:
5485:
5484:
5473:
5470:
5467:
5464:
5461:
5456:
5453:
5449:
5445:
5440:
5437:
5433:
5410:
5409:
5398:
5395:
5392:
5389:
5386:
5383:
5378:
5373:
5369:
5365:
5360:
5355:
5351:
5347:
5342:
5339:
5334:
5329:
5306:
5305:
5294:
5291:
5288:
5285:
5282:
5277:
5274:
5270:
5266:
5261:
5258:
5254:
5230:
5227:
5208:
5204:
5181:
5177:
5153:
5148:
5144:
5140:
5135:
5131:
5127:
5122:
5118:
5114:
5109:
5105:
5101:
5098:
5093:
5090:
5086:
5065:
5062:
5059:
5056:
5051:
5048:
5044:
5021:
5018:
5015:
5010:
5006:
5002:
4997:
4993:
4989:
4986:
4981:
4977:
4971:
4967:
4963:
4958:
4954:
4948:
4944:
4932:Pauli matrices
4917:
4913:
4892:
4887:
4883:
4879:
4876:
4873:
4870:
4864:
4861:
4857:
4852:
4847:
4843:
4839:
4836:
4833:
4830:
4824:
4821:
4817:
4812:
4809:
4804:
4798:
4795:
4792:
4786:
4783:
4779:
4774:
4772:
4769:
4766:
4760:
4757:
4753:
4748:
4747:
4744:
4741:
4738:
4732:
4729:
4725:
4720:
4717:
4715:
4712:
4709:
4703:
4700:
4696:
4691:
4688:
4687:
4685:
4680:
4675:
4671:
4651:
4646:
4642:
4638:
4635:
4632:
4627:
4623:
4619:
4614:
4610:
4604:
4601:
4596:
4591:
4585:
4582:
4579:
4576:
4572:
4568:
4562:
4559:
4554:
4553:
4548:
4544:
4538:
4535:
4530:
4528:
4525:
4522:
4521:
4519:
4514:
4509:
4505:
4482:
4478:
4455:
4450:
4428:
4425:
4422:
4419:
4416:
4391:
4386:
4360:
4357:
4354:
4349:
4346:
4328:
4325:
4309:
4306:
4303:
4277:
4273:
4270:
4267:
4264:
4261:
4258:
4255:
4250:
4246:
4242:
4239:
4234:
4229:
4225:
4221:
4216:
4211:
4207:
4203:
4198:
4195:
4189:
4185:
4182:
4176:
4171:
4167:
4164:
4144:
4135:of a solution
4120:
4116:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4057:
4050:
4047:
4043:
4038:
4035:
4032:
4026:
4021:
4014:
4011:
4007:
4002:
3999:
3979:
3970:of a solution
3968:winding number
3959:
3956:
3943:
3923:
3903:
3900:
3897:
3877:
3851:
3831:
3811:
3796:
3795:
3776:
3769:
3765:
3762:
3759:
3751:
3746:
3743:
3738:
3735:
3730:
3725:
3721:
3717:
3714:
3711:
3709:
3705:
3701:
3697:
3696:
3691:
3684:
3680:
3677:
3674:
3666:
3661:
3658:
3655:
3652:
3649:
3644:
3640:
3636:
3633:
3631:
3627:
3623:
3619:
3618:
3604:
3603:
3591:
3588:
3585:
3582:
3579:
3574:
3571:
3567:
3543:
3525:
3522:
3519:
3518:
3500:
3461:
3457:
3452:
3448:
3425:
3421:
3400:
3374:
3362:
3361:
3350:
3342:
3338:
3334:
3331:
3324:
3319:
3310:
3306:
3303:
3300:
3297:
3292:
3288:
3284:
3281:
3278:
3273:
3270:
3267:
3261:
3252:
3222:
3208:
3205:
3202:
3201:
3183:
3182:
3169:
3141:
3137:
3131:
3128:
3120:
3116:
3112:
3109:
3104:
3101:
3098:
3094:
3089:
3086:
3083:
3080:
3077:
3074:
3066:
3062:
3058:
3055:
3047:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
2982:
2981:
2968:
2940:
2929:
2925:
2921:
2918:
2913:
2910:
2904:
2901:
2891:
2887:
2883:
2880:
2876:
2871:
2868:
2865:
2859:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2826:
2823:
2819:
2815:
2811:
2787:
2776:
2772:
2768:
2765:
2760:
2757:
2751:
2748:
2738:
2734:
2730:
2727:
2723:
2718:
2715:
2712:
2706:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2673:
2669:
2665:
2661:
2616:
2613:
2598:
2594:
2590:
2587:
2584:
2567:
2566:
2552:
2523:
2520:
2517:
2508:
2483:
2480:
2477:
2468:
2451:for all time.
2449:
2448:
2437:
2434:
2429:
2425:
2421:
2418:
2408:
2404:
2401:
2397:
2394:
2387:
2384:
2379:
2376:
2371:
2366:
2362:
2358:
2355:
2351:
2347:
2343:
2331:
2330:
2319:
2314:
2310:
2307:
2303:
2300:
2293:
2290:
2287:
2284:
2281:
2276:
2272:
2268:
2264:
2260:
2256:
2224:
2204:
2201:
2198:
2195:
2192:
2172:
2169:
2166:
2163:
2143:
2140:
2137:
2117:
2093:
2082:
2081:
2070:
2067:
2064:
2061:
2058:
2053:
2049:
2045:
2040:
2037:
2033:
2029:
2024:
2021:
2017:
2002:
2001:
1990:
1982:
1978:
1974:
1971:
1967:
1962:
1957:
1953:
1938:
1937:
1926:
1922:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1886:
1882:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1845:
1825:
1822:
1813:
1810:
1809:
1808:
1793:
1787:
1784:
1781:
1778:
1775:
1768:
1764:
1758:
1754:
1750:
1747:
1739:
1734:
1731:
1728:
1724:
1720:
1717:
1714:
1711:
1700:
1694:
1691:
1689:
1687:
1681:
1678:
1675:
1672:
1669:
1662:
1658:
1652:
1648:
1644:
1641:
1633:
1628:
1625:
1622:
1618:
1614:
1609:
1604:
1600:
1594:
1591:
1586:
1581:
1577:
1573:
1568:
1563:
1559:
1555:
1550:
1547:
1542:
1539:
1537:
1535:
1532:
1529:
1518:
1512:
1511:
1493:
1492:
1481:
1475:
1472:
1469:
1466:
1463:
1456:
1452:
1446:
1442:
1438:
1435:
1427:
1422:
1419:
1416:
1412:
1408:
1405:
1402:
1399:
1396:
1393:
1371:
1370:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1333:
1328:
1324:
1320:
1315:
1310:
1306:
1302:
1297:
1294:
1289:
1286:
1283:
1280:
1269:
1244:
1243:
1232:
1229:
1226:
1223:
1218:
1215:
1211:
1207:
1202:
1199:
1195:
1175:
1172:
1157:
1153:
1149:
1146:
1142:
1136:
1131:
1127:
1123:
1120:
1117:
1112:
1109:
1105:
1101:
1079:
1076:
1072:
1068:
1037:
1033:
1030:
1027:
1024:
1021:
1014:
1003:
997:
994:
990:
986:
979:
968:
962:
959:
955:
951:
927:
907:
883:
880:
875:Main article:
872:
869:
856:
836:
833:
830:
827:
824:
804:
789:Lorentz boosts
783:introduced by
764:
761:
711:
708:
705:
702:
699:
694:
691:
687:
662:
659:
656:
653:
650:
647:
644:
641:
638:
635:
632:
617:
616:
605:
600:
596:
592:
589:
586:
583:
579:
576:
572:
569:
566:
563:
560:
555:
551:
547:
544:
539:
535:
531:
501:
456:
455:
444:
441:
438:
435:
432:
427:
424:
420:
405:
404:
393:
388:
384:
381:
378:
372:
369:
365:
360:
356:
353:
350:
344:
341:
311:
310:
299:
296:
293:
290:
287:
284:
281:
276:
273:
269:
265:
260:
257:
253:
229:
226:
223:
220:
217:
194:
191:
189:
186:
174:integrable PDE
115:
87:
67:
47:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
9237:
9226:
9223:
9221:
9218:
9216:
9213:
9211:
9208:
9206:
9203:
9201:
9198:
9197:
9195:
9176:
9173:
9171:
9168:
9166:
9163:
9161:
9160:Rodney Baxter
9158:
9157:
9155:
9151:
9145:
9142:
9140:
9137:
9136:
9134:
9130:
9124:
9121:
9119:
9116:
9114:
9111:
9109:
9106:
9104:
9101:
9100:
9098:
9094:
9088:
9085:
9083:
9080:
9078:
9075:
9073:
9070:
9068:
9067:Nigel Hitchin
9065:
9063:
9060:
9058:
9055:
9053:
9050:
9049:
9047:
9043:
9040:
9036:
9026:
9023:
9021:
9018:
9016:
9012:
9011:
9009:
9005:
8999:
8996:
8994:
8991:
8990:
8988:
8984:
8981:
8977:
8971:
8968:
8966:
8963:
8961:
8958:
8956:
8953:
8950:
8947:
8946:
8944:
8942:
8937:
8927:
8924:
8922:(Hamiltonian)
8921:
8918:
8915:
8912:
8911:
8909:
8905:
8899:
8896:
8894:
8891:
8889:
8886:
8884:
8881:
8879:
8876:
8874:
8871:
8870:
8868:
8864:
8854:
8851:
8849:
8846:
8844:
8841:
8840:
8838:
8834:
8828:
8825:
8821:
8818:
8817:
8816:
8813:
8811:
8807:
8805:
8802:
8800:
8797:
8796:
8794:
8790:
8784:
8781:
8779:
8776:
8774:
8771:
8769:
8766:
8764:
8761:
8757:
8756:KdV hierarchy
8754:
8753:
8752:
8749:
8748:
8746:
8742:
8739:
8735:
8729:
8726:
8724:
8723:Hydrogen atom
8721:
8719:
8716:
8715:
8713:
8709:
8699:
8696:
8694:
8691:
8689:
8686:
8685:
8683:
8679:
8673:
8670:
8668:
8665:
8661:
8658:
8656:
8653:
8651:
8648:
8647:
8646:
8643:
8639:
8636:
8634:
8633:Kepler system
8631:
8630:
8629:
8626:
8624:
8621:
8620:
8618:
8614:
8611:
8607:
8601:
8598:
8594:
8591:
8590:
8589:
8586:
8585:
8583:
8579:
8575:
8568:
8563:
8561:
8556:
8554:
8549:
8548:
8545:
8535:
8527:
8522:
8516:
8513:
8511:
8510:Quantum logic
8508:
8504:
8501:
8500:
8499:
8496:
8494:
8491:
8489:
8486:
8484:
8481:
8477:
8474:
8473:
8472:
8469:
8465:
8462:
8461:
8460:
8457:
8455:
8452:
8450:
8447:
8445:
8444:Quantum chaos
8442:
8440:
8437:
8435:
8432:
8430:
8427:
8425:
8422:
8420:
8419:Cosmic string
8417:
8415:
8412:
8411:
8409:
8405:
8395:
8392:
8390:
8387:
8385:
8382:
8380:
8377:
8375:
8372:
8370:
8367:
8365:
8362:
8361:
8359:
8355:
8349:
8346:
8344:
8341:
8340:
8338:
8334:
8328:
8325:
8323:
8320:
8318:
8315:
8313:
8310:
8308:
8305:
8303:
8300:
8298:
8295:
8293:
8292:Pure 4D N = 1
8290:
8289:
8287:
8283:
8277:
8274:
8272:
8269:
8267:
8264:
8263:
8261:
8257:
8251:
8248:
8246:
8243:
8241:
8238:
8236:
8233:
8231:
8228:
8227:
8225:
8221:
8215:
8212:
8210:
8207:
8205:
8202:
8200:
8197:
8195:
8192:
8191:
8189:
8185:
8179:
8176:
8174:
8173:ThirringâWess
8171:
8169:
8166:
8164:
8161:
8159:
8156:
8154:
8151:
8149:
8148:BulloughâDodd
8146:
8144:
8143:2D YangâMills
8141:
8140:
8138:
8134:
8128:
8125:
8123:
8120:
8118:
8115:
8113:
8110:
8108:
8105:
8103:
8100:
8098:
8095:
8093:
8090:
8088:
8085:
8083:
8080:
8078:
8075:
8073:
8070:
8068:
8065:
8063:
8060:
8059:
8057:
8053:
8050:
8046:
8040:
8037:
8035:
8032:
8030:
8027:
8025:
8022:
8020:
8019:String theory
8017:
8015:
8012:
8010:
8007:
8005:
8002:
8000:
7997:
7995:
7992:
7990:
7989:Axiomatic QFT
7987:
7985:
7984:Algebraic QFT
7982:
7981:
7979:
7975:
7971:
7964:
7959:
7957:
7952:
7950:
7945:
7944:
7941:
7934:
7930:
7927:
7924:
7921:
7918:
7915:
7912:
7911:
7907:
7898:
7893:
7886:
7883:
7878:
7874:
7870:
7866:
7862:
7858:
7851:
7848:
7842:
7837:
7833:
7829:
7825:
7821:
7817:
7810:
7807:
7802:
7798:
7794:
7790:
7786:
7782:
7778:
7774:
7770:
7763:
7760:
7755:
7751:
7747:
7743:
7739:
7735:
7728:
7725:
7713:
7707:
7703:
7702:
7694:
7691:
7686:
7682:
7678:
7674:
7670:
7666:
7661:
7656:
7652:
7648:
7641:
7639:
7635:
7623:
7619:
7615:
7611:
7607:
7603:
7598:
7593:
7589:
7585:
7581:
7574:
7571:
7565:
7560:
7553:
7550:
7544:
7539:
7532:
7529:
7517:
7513:
7509:
7505:
7501:
7497:
7493:
7486:
7483:
7478:
7474:
7470:
7466:
7462:
7458:
7451:
7448:
7443:
7439:
7435:
7431:
7427:
7423:
7416:
7413:
7408:
7404:
7400:
7396:
7392:
7388:
7381:
7378:
7373:
7369:
7365:
7361:
7357:
7353:
7346:
7343:
7338:
7334:
7330:
7326:
7322:
7318:
7314:
7310:
7303:
7300:
7295:
7291:
7287:
7283:
7279:
7275:
7270:
7265:
7261:
7257:
7250:
7248:
7246:
7242:
7237:
7233:
7229:
7225:
7218:
7215:
7210:
7206:
7202:
7198:
7194:
7190:
7183:
7180:
7175:
7169:
7165:
7158:
7155:
7150:
7144:
7140:
7133:
7130:
7126:
7125:
7118:
7116:
7112:
7107:
7101:
7097:
7093:
7086:
7083:
7077:
7072:
7068:
7064:
7060:
7053:
7051:
7049:
7047:
7045:
7043:
7041:
7039:
7037:
7033:
7028:
7024:
7020:
7016:
7012:
7008:
7001:
6998:
6988:
6982:
6978:
6974:
6970:
6966:
6959:
6956:
6951:
6945:
6937:
6933:
6926:
6919:
6916:
6911:
6905:
6901:
6894:
6891:
6886:
6880:
6876:
6869:
6867:
6863:
6858:
6854:
6850:
6846:
6842:
6838:
6831:
6828:
6823:
6819:
6812:
6809:
6804:
6800:
6797:(39): 1â148.
6796:
6792:
6788:
6781:
6778:
6772:
6768:
6765:
6763:
6760:
6758:
6755:
6754:
6750:
6748:
6746:
6741:
6739:
6735:
6731:
6727:
6723:
6719:
6715:
6710:
6708:
6703:
6701:
6693:
6691:
6685:
6683:
6669:
6666:
6660:
6657:
6654:
6650:
6645:
6640:
6636:
6627:
6611:
6591:
6588:
6585:
6582:
6579:
6559:
6556:
6553:
6547:
6544:
6541:
6538:
6532:
6529:
6526:
6523:
6520:
6512:
6509:
6504:
6501:
6496:
6482:
6480:
6479:Martin Hairer
6476:
6472:
6464:
6462:
6460:
6440:
6437:
6434:
6429:
6425:
6404:
6401:
6398:
6393:
6389:
6380:
6364:
6361:
6358:
6353:
6349:
6328:
6325:
6319:
6316:
6313:
6309:
6287:
6284:
6281:
6276:
6272:
6268:
6265:
6262:
6254:
6250:
6234:
6231:
6228:
6223:
6219:
6210:
6209:finite regime
6205:
6203:
6202:JĂŒrg Fröhlich
6199:
6181:
6177:
6164:
6162:
6160:
6144:
6139:
6136:
6133:
6129:
6125:
6120:
6116:
6087:
6084:
6080:
6076:
6071:
6067:
6060:
6057:
6052:
6048:
6042:
6037:
6033:
6027:
6024:
6019:
6016:
6011:
6003:
5998:
5988:
5985:
5980:
5975:
5972:
5969:
5953:
5952:
5951:
5950:
5945:
5943:
5939:
5921:
5918:
5913:
5910:
5890:
5868:
5864:
5841:
5837:
5814:
5810:
5789:
5786:
5781:
5777:
5768:
5764:
5760:
5756:
5752:
5748:
5744:
5740:
5735:
5733:
5729:
5725:
5717:
5715:
5713:
5704:
5702:
5686:
5660:
5642:
5623:
5619:
5614:
5601:
5596:
5593:
5589:
5585:
5582:
5577:
5574:
5570:
5566:
5561:
5558:
5554:
5545:
5544:
5539:
5537:
5532:
5530:
5526:
5522:
5521:Wick rotation
5518:
5514:
5510:
5494:
5471:
5468:
5465:
5462:
5459:
5454:
5451:
5447:
5443:
5438:
5435:
5431:
5423:
5422:
5421:
5419:
5415:
5396:
5393:
5390:
5387:
5384:
5376:
5371:
5367:
5363:
5358:
5353:
5349:
5340:
5337:
5332:
5318:
5317:
5316:
5315:
5311:
5292:
5289:
5286:
5283:
5280:
5275:
5272:
5268:
5264:
5259:
5256:
5252:
5244:
5243:
5242:
5240:
5228:
5226:
5224:
5206:
5202:
5179:
5175:
5165:
5146:
5142:
5138:
5133:
5125:
5120:
5116:
5112:
5107:
5096:
5091:
5088:
5084:
5063:
5060:
5057:
5054:
5049:
5046:
5042:
5032:
5019:
5016:
5008:
5004:
5000:
4995:
4991:
4984:
4979:
4975:
4969:
4961:
4956:
4952:
4946:
4933:
4915:
4911:
4890:
4885:
4881:
4877:
4874:
4871:
4868:
4862:
4859:
4855:
4850:
4845:
4841:
4837:
4834:
4831:
4828:
4822:
4819:
4815:
4810:
4807:
4802:
4796:
4793:
4790:
4784:
4781:
4777:
4770:
4767:
4764:
4758:
4755:
4751:
4742:
4739:
4736:
4730:
4727:
4723:
4718:
4713:
4710:
4707:
4701:
4698:
4694:
4689:
4683:
4678:
4673:
4669:
4649:
4644:
4640:
4636:
4633:
4630:
4625:
4621:
4617:
4612:
4608:
4602:
4599:
4594:
4589:
4583:
4580:
4577:
4570:
4566:
4560:
4557:
4546:
4542:
4536:
4533:
4526:
4523:
4517:
4512:
4507:
4503:
4495:are given by
4480:
4476:
4453:
4423:
4420:
4417:
4405:
4389:
4374:
4355:
4334:
4326:
4324:
4321:
4307:
4304:
4301:
4291:
4275:
4268:
4265:
4262:
4259:
4256:
4248:
4244:
4240:
4232:
4227:
4223:
4219:
4214:
4209:
4205:
4196:
4193:
4187:
4183:
4180:
4169:
4165:
4162:
4142:
4134:
4118:
4114:
4107:
4104:
4098:
4095:
4092:
4086:
4083:
4077:
4074:
4068:
4065:
4059:
4055:
4048:
4045:
4041:
4036:
4033:
4030:
4019:
4012:
4009:
4005:
4000:
3997:
3977:
3969:
3965:
3957:
3955:
3941:
3921:
3901:
3898:
3895:
3875:
3866:
3863:
3849:
3829:
3809:
3801:
3767:
3763:
3760:
3757:
3744:
3741:
3736:
3733:
3728:
3723:
3719:
3715:
3712:
3710:
3703:
3699:
3682:
3678:
3675:
3672:
3659:
3656:
3653:
3650:
3647:
3642:
3638:
3634:
3632:
3625:
3621:
3609:
3608:
3607:
3589:
3586:
3583:
3580:
3577:
3572:
3569:
3565:
3557:
3556:
3555:
3541:
3534:Suppose that
3531:
3523:
3514:
3510:
3507:Collision of
3505:
3501:
3496:
3492:
3489:Collision of
3487:
3483:
3482:
3479:
3455:
3450:
3446:
3423:
3419:
3398:
3372:
3348:
3340:
3336:
3332:
3329:
3317:
3308:
3304:
3301:
3290:
3286:
3282:
3279:
3271:
3268:
3265:
3259:
3242:
3241:
3240:
3206:
3197:
3193:
3189:
3188:
3178:
3174:
3170:
3165:
3160:
3156:
3155:
3152:
3139:
3135:
3126:
3118:
3114:
3110:
3107:
3099:
3096:
3092:
3084:
3081:
3075:
3072:
3064:
3060:
3056:
3053:
3045:
3041:
3038:
3035:
3032:
3026:
3023:
3020:
3014:
3005:
3003:
2999:
2995:
2991:
2990:
2977:
2973:
2969:
2964:
2963:Antikink-kink
2960:
2956:
2955:
2952:
2938:
2927:
2923:
2919:
2916:
2911:
2908:
2902:
2899:
2889:
2885:
2881:
2878:
2874:
2869:
2866:
2863:
2857:
2853:
2850:
2847:
2844:
2838:
2835:
2832:
2824:
2821:
2817:
2813:
2809:
2799:
2785:
2774:
2770:
2766:
2763:
2758:
2755:
2749:
2746:
2736:
2732:
2728:
2725:
2721:
2716:
2713:
2710:
2704:
2700:
2697:
2694:
2691:
2685:
2682:
2679:
2671:
2667:
2663:
2659:
2649:
2647:
2643:
2639:
2635:
2630:
2626:
2622:
2614:
2596:
2592:
2588:
2585:
2582:
2573:
2562:
2557:
2553:
2548:
2543:
2539:
2538:
2535:
2521:
2518:
2515:
2506:
2497:
2481:
2478:
2475:
2466:
2457:
2452:
2435:
2432:
2427:
2423:
2419:
2416:
2406:
2402:
2399:
2395:
2392:
2385:
2382:
2377:
2374:
2369:
2364:
2360:
2356:
2353:
2349:
2345:
2341:
2333:
2332:
2317:
2312:
2308:
2305:
2301:
2298:
2291:
2288:
2285:
2282:
2279:
2274:
2270:
2266:
2262:
2258:
2254:
2246:
2245:
2244:
2242:
2238:
2235:is called an
2222:
2202:
2199:
2196:
2193:
2190:
2183:. The states
2170:
2167:
2164:
2161:
2141:
2138:
2135:
2115:
2107:
2091:
2068:
2065:
2062:
2059:
2056:
2051:
2047:
2043:
2038:
2035:
2031:
2027:
2022:
2019:
2015:
2007:
2006:
2005:
1988:
1980:
1976:
1972:
1969:
1965:
1960:
1955:
1951:
1943:
1942:
1941:
1924:
1920:
1915:
1912:
1906:
1903:
1900:
1897:
1891:
1888:
1884:
1880:
1876:
1873:
1870:
1867:
1861:
1858:
1855:
1843:
1835:
1834:
1833:
1831:
1823:
1821:
1819:
1811:
1791:
1785:
1779:
1776:
1766:
1756:
1752:
1748:
1732:
1729:
1726:
1722:
1718:
1712:
1692:
1690:
1679:
1673:
1670:
1660:
1650:
1646:
1642:
1626:
1623:
1620:
1616:
1612:
1607:
1602:
1598:
1592:
1584:
1579:
1575:
1571:
1566:
1561:
1557:
1548:
1545:
1540:
1538:
1530:
1502:
1501:
1500:
1498:
1479:
1473:
1467:
1464:
1454:
1444:
1440:
1436:
1420:
1417:
1414:
1410:
1406:
1400:
1394:
1391:
1384:
1383:
1382:
1380:
1376:
1375:Taylor series
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1331:
1326:
1322:
1318:
1313:
1308:
1304:
1295:
1292:
1287:
1281:
1257:
1256:
1255:
1253:
1249:
1230:
1227:
1224:
1221:
1216:
1213:
1209:
1205:
1200:
1197:
1193:
1185:
1184:
1183:
1181:
1173:
1171:
1155:
1151:
1147:
1144:
1134:
1129:
1125:
1121:
1118:
1110:
1107:
1103:
1099:
1077:
1074:
1070:
1066:
1057:
1035:
1031:
1028:
1025:
1022:
1019:
1012:
1001:
995:
992:
988:
984:
977:
966:
960:
957:
953:
949:
925:
905:
893:
888:
881:
878:
870:
868:
854:
834:
831:
828:
825:
822:
802:
792:
790:
786:
782:
781:Lie transform
778:
774:
770:
762:
758:
753:
749:
747:
743:
738:
736:
733:
729:
725:
709:
706:
703:
700:
697:
692:
689:
685:
676:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
622:
603:
598:
594:
590:
587:
584:
581:
577:
574:
570:
567:
564:
561:
558:
553:
549:
545:
542:
537:
533:
529:
522:
521:
520:
518:
513:
499:
491:
487:
483:
479:
474:
472:
468:
465:
461:
442:
439:
436:
433:
430:
425:
422:
418:
410:
409:
408:
391:
386:
382:
379:
376:
370:
367:
363:
358:
354:
351:
348:
342:
339:
332:
331:
330:
328:
324:
320:
316:
297:
294:
291:
288:
285:
282:
279:
274:
271:
267:
263:
258:
255:
251:
243:
242:
241:
224:
221:
218:
207:
203:
198:
192:
187:
185:
183:
179:
175:
171:
166:
164:
161:known as the
160:
156:
152:
148:
144:
140:
136:
132:
127:
113:
105:
101:
100:wave operator
85:
65:
45:
37:
33:
19:
9123:Robert Miura
9038:Contributors
9015:Bethe ansatz
8998:Gaudin model
8916:(Lagrangian)
8762:
8751:KdV equation
8655:Kovalevskaya
8525:
8454:Quantum foam
8394:Stueckelberg
8348:ChernâSimons
8285:Supergravity
8162:
8024:Supergravity
8009:Gauge theory
7885:
7860:
7856:
7850:
7823:
7819:
7809:
7776:
7772:
7762:
7737:
7733:
7727:
7715:. Retrieved
7700:
7693:
7650:
7646:
7625:. Retrieved
7587:
7583:
7573:
7552:
7531:
7519:. Retrieved
7499:
7495:
7485:
7460:
7456:
7450:
7425:
7421:
7415:
7390:
7386:
7380:
7355:
7351:
7345:
7312:
7308:
7302:
7259:
7255:
7227:
7223:
7217:
7192:
7188:
7182:
7163:
7157:
7138:
7132:
7122:
7091:
7085:
7066:
7062:
7010:
7006:
7000:
6990:, retrieved
6968:
6958:
6944:cite journal
6935:
6931:
6918:
6899:
6893:
6874:
6840:
6836:
6830:
6821:
6817:
6811:
6794:
6790:
6780:
6742:
6711:
6705:Dynamics in
6704:
6697:
6689:
6483:
6474:
6470:
6468:
6341:passed. For
6252:
6249:counterterms
6208:
6206:
6168:
6107:
5946:
5937:
5736:
5721:
5711:
5708:
5615:
5546:
5540:
5535:
5533:
5528:
5524:
5512:
5508:
5486:
5417:
5413:
5411:
5308:This is the
5307:
5241:is given by
5234:
5232:
5166:
5033:
4406:
4330:
4322:
4292:
4132:
3967:
3963:
3961:
3867:
3864:
3799:
3797:
3605:
3533:
3512:
3508:
3494:
3490:
3363:
3239:is given by
3210:
3195:
3176:
3163:
3006:
3001:
2997:
2993:
2987:
2985:
2975:
2962:
2800:
2650:
2618:
2560:
2546:
2496:right-handed
2453:
2450:
2236:
2105:
2104:is called a
2083:
2003:
1939:
1827:
1815:
1494:
1372:
1254:is given by
1245:
1182:in physics:
1177:
1058:
897:
891:
867:an integer.
793:
766:
757:Dini surface
745:
739:
735:pseudosphere
731:
618:
514:
485:
481:
475:
466:
462:of constant
457:
406:
326:
322:
318:
312:
205:
199:
196:
178:relativistic
177:
167:
128:
31:
29:
8949:Ising model
8878:Quantum KdV
8336:Topological
8250:WessâZumino
8163:Sine-Gordon
8153:GrossâNeveu
8062:BornâInfeld
8029:Thermal QFT
7428:(1): 1â87.
6938:(1): 17â25.
6767:Shape waves
6722:Coulomb gas
6626:white noise
6247:, where no
5420:, given by
3491:moving kink
2634:phase shift
2456:left-handed
1832:solutions:
726:due to the
325:), akin to
18:Sine-Gordon
9194:Categories
9013:Algebraic
8117:YangâMills
7564:1808.02594
7543:2205.09223
7521:27 January
7262:(3): 047.
6992:2023-11-17
6824:: 137â149.
6773:References
6471:stochastic
6459:subalgebra
5314:Lagrangian
4903:where the
4373:connection
3528:See also:
2978:collision.
2965:collision.
2559:Traveling
2545:Traveling
1373:Using the
785:Sophus Lie
490:arc length
208:, denoted
9118:Peter Lax
8804:Lax pairs
8593:Foliation
8526:See also:
8245:Super QCD
8199:Liouville
8187:Conformal
8158:Schwinger
7801:120798940
7717:22 August
7622:253750515
7597:1409.5724
7337:121527379
7294:119501952
7069:: 67â75.
6670:π
6637:β
6612:ξ
6592:θ
6586:β
6557:ξ
6548:θ
6539:β
6533:
6518:Δ
6493:∂
6455:affine sl
6441:π
6426:β
6405:π
6390:β
6365:π
6350:β
6329:π
6288:π
6273:β
6266:π
6235:π
6220:β
6178:β
6140:φ
6137:β
6121:β
6088:β
6085:−
6072:β
6061:α
6058:−
6049:φ
6017:φ
6012:μ
6008:∂
6004:φ
5999:μ
5995:∂
5922:π
5911:β
5891:β
5865:γ
5838:α
5811:γ
5790:β
5778:α
5732:breathers
5680:^
5597:φ
5571:φ
5567:−
5555:φ
5495:φ
5469:φ
5466:
5448:φ
5432:φ
5394:φ
5391:
5385:−
5368:φ
5364:−
5350:φ
5290:φ
5287:
5269:φ
5265:−
5253:φ
5147:ν
5139:−
5134:ν
5130:∂
5121:μ
5113:−
5108:μ
5104:∂
5092:ν
5089:μ
5064:φ
5061:
5043:φ
4966:∂
4962:−
4943:∂
4912:σ
4882:σ
4878:φ
4875:
4863:λ
4851:−
4842:σ
4838:φ
4835:
4823:λ
4811:−
4797:φ
4794:
4785:λ
4771:φ
4768:
4759:λ
4743:φ
4740:
4731:λ
4719:−
4714:φ
4711:
4702:λ
4690:−
4641:σ
4634:λ
4622:σ
4609:φ
4584:λ
4578:−
4567:φ
4543:φ
4527:λ
4481:μ
4333:curvature
4269:φ
4266:
4260:−
4224:φ
4206:φ
4170:∫
4143:φ
4102:∞
4099:−
4087:φ
4084:−
4072:∞
4060:φ
4049:π
4034:φ
4020:∫
4013:π
3978:φ
3922:ψ
3899:≡
3896:φ
3876:φ
3850:ψ
3830:φ
3810:ψ
3764:φ
3761:−
3758:ψ
3745:
3720:φ
3716:−
3700:ψ
3679:φ
3673:ψ
3660:
3639:φ
3622:ψ
3587:φ
3584:
3566:φ
3542:φ
3460:Δ
3399:ω
3337:ω
3333:−
3305:−
3287:ω
3283:−
3272:
3251:Δ
3221:Δ
3115:ω
3111:−
3100:
3093:ω
3082:ω
3076:
3061:ω
3057:−
3042:
3015:φ
2976:Kink-kink
2920:−
2903:
2882:−
2870:
2854:
2810:φ
2767:−
2750:
2729:−
2717:
2701:
2660:φ
2589:
2507:θ
2479:−
2467:θ
2424:φ
2417:φ
2403:φ
2400:−
2393:φ
2386:
2378:β
2361:φ
2357:−
2342:φ
2309:φ
2299:φ
2292:
2286:β
2271:φ
2255:φ
2223:γ
2200:π
2194:≅
2191:φ
2171:π
2162:φ
2136:φ
2116:φ
2092:γ
2063:φ
2060:
2032:φ
2028:−
2016:φ
1973:−
1952:γ
1916:δ
1901:−
1892:γ
1877:
1844:φ
1753:φ
1749:−
1738:∞
1723:∑
1713:φ
1647:φ
1643:−
1632:∞
1617:∑
1599:φ
1593:−
1576:φ
1572:−
1558:φ
1531:φ
1441:φ
1437:−
1426:∞
1411:∑
1401:φ
1395:
1355:φ
1352:
1340:−
1323:φ
1319:−
1305:φ
1282:φ
1225:φ
1210:φ
1206:−
1194:φ
1145:−
1126:φ
1104:φ
1071:φ
1036:⏟
1032:φ
1029:
1013:−
1002:⏟
989:φ
967:⏟
954:φ
926:φ
835:π
823:φ
803:φ
707:φ
704:
686:φ
661:φ
658:
571:φ
568:
500:φ
440:φ
437:
419:φ
380:−
289:φ
286:
268:φ
264:−
252:φ
114:φ
46:φ
9210:Surfaces
9200:Solitons
8986:Examples
8744:Examples
8660:Lagrange
8616:Examples
8322:Type IIB
8317:Type IIA
8302:4D N = 8
8297:4D N = 1
8266:6D (2,0)
8230:4D N = 1
8209:Polyakov
8168:Thirring
7977:Theories
7929:Archived
7685:18230581
6803:55567842
6751:See also
6726:vortices
5940:massive
5223:Lax pair
4930:are the
2989:breather
2638:velocity
2561:antikink
2396:′
2350:′
2302:′
2263:′
2237:antikink
773:BĂ€cklund
724:singular
673:and the
619:and the
460:surfaces
102:and the
8424:History
8407:Related
8204:Minimal
8055:Regular
7865:Bibcode
7828:Bibcode
7781:Bibcode
7742:Bibcode
7665:Bibcode
7602:Bibcode
7504:Bibcode
7465:Bibcode
7430:Bibcode
7395:Bibcode
7360:Bibcode
7317:Bibcode
7274:Bibcode
7197:Bibcode
7015:Bibcode
6845:Bibcode
6716:as the
5763:Coleman
5757:to the
5312:of the
3914:, then
2621:soliton
1848:soliton
1830:soliton
1818:soliton
1042:gravity
1008:tension
769:Bianchi
321:,
170:soliton
141:as the
133: (
9007:Theory
8792:Theory
8681:Theory
8364:Chiral
8312:Type I
8127:Yukawa
8048:Models
7799:
7708:
7683:
7627:14 May
7620:
7335:
7292:
7170:
7145:
7102:
6983:
6906:
6881:
6801:
6762:Fluxon
6720:for a
6572:where
6196:, the
5755:S-dual
5487:where
4133:energy
3798:where
3364:where
3269:artanh
3039:arctan
3000:, and
2851:arctan
2698:arctan
2619:Multi-
2586:arctan
1940:where
1874:arctan
1379:cosine
1174:Naming
329:where
149:â1 in
9132:IQFTs
8650:Euler
8503:links
8476:links
8464:links
8384:NMSSM
8369:Fermi
8112:Soler
8082:Proca
7892:arXiv
7797:S2CID
7681:S2CID
7655:arXiv
7618:S2CID
7592:arXiv
7559:arXiv
7538:arXiv
7333:S2CID
7290:S2CID
7264:arXiv
6928:(PDF)
6108:with
2642:shape
9096:PDEs
8374:MSSM
8271:ABJM
8178:Toda
7719:2023
7706:ISBN
7629:2023
7523:2023
7260:2007
7168:ISBN
7143:ISBN
7100:ISBN
6981:ISBN
6950:link
6904:ISBN
6879:ISBN
6799:OCLC
6469:The
6417:and
6379:1975
6359:>
6282:<
6269:<
6229:<
6207:The
5938:free
5802:and
5741:and
5534:The
5519:(or
5511:and
5388:cosh
5284:sinh
5233:The
5194:and
4131:The
3962:The
3842:and
3511:and
3493:and
3162:The
3097:cosh
2900:sinh
2867:cosh
2747:cosh
2714:sinh
2640:and
2547:kink
2106:kink
847:for
771:and
515:The
202:real
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