655:
461:
467:
956:
157:
may be parametrised by only finitely many of their dependent variables, in this case 2 of them. It was first introduced by John
Gibbons and Serguei Tsarev in 1996, This system was also derived, as a condition that two quadratic Hamiltonians should have vanishing
332:
650:{\displaystyle {\frac {\partial A^{0}}{\partial \lambda _{i}\partial \lambda _{j}}}=2{\frac {{\frac {\partial A^{0}}{\partial \lambda _{i}}}{\frac {\partial A^{0}}{\lambda _{j}}}}{(p_{i}-p_{j})^{2}}}.\qquad (2b)}
1302:
844:
144:
1049:
1136:
1209:
805:
778:
297:
1431:
731:
1176:
684:
1359:
1332:
1079:
832:
324:
266:
235:
751:
711:
208:
188:
689:
This system has solutions parametrised by N functions of a single variable. A class of these may be constructed in terms of N-parameter families of
456:{\displaystyle {\frac {\partial p_{i}}{\partial \lambda _{j}}}=-{\frac {\frac {\partial A^{0}}{\partial \lambda _{j}}}{p_{i}-p_{j}}},\qquad (2a)}
1217:
951:{\displaystyle {\frac {\partial p}{\partial \lambda _{i}}}=-{\frac {\frac {\partial A^{0}}{\partial \lambda _{j}}}{p-p_{i}}}.\qquad (3)}
1416:
J. Gibbons and S.P. Tsarev, Conformal Maps and the reduction of Benney equations, Phys
Letters A, vol 258, No4-6, pp 263–271, 1999.
45:
972:
835:
36:
1389:
J. Gibbons and S.P. Tsarev, Reductions of the Benney
Equations, Physics Letters A, Vol. 211, Issue 1, Pages 19–24, 1996.
150:
1087:
154:
1181:
733:-plane but with N slits. Each slit is taken along a fixed curve with one end fixed on the boundary of
783:
756:
275:
716:
269:
238:
32:
1148:
663:
1337:
1310:
1057:
810:
302:
244:
213:
966:
An elementary family of solutions to the N-dimensional problem may be derived by setting:
159:
834:. The system can then be understood as the consistency condition between the set of N
736:
696:
193:
173:
17:
1425:
690:
190:
independent variables, one looks for solutions of the Benney hierarchy in which only
170:
The theory of this equation was subsequently developed by
Gibbons and Tsarev. In
1142:
39:. In its simplest form, in two dimensions, it may be written as follows:
1297:{\displaystyle A^{0}={\frac {1}{N+1}}\sum \sum _{i>j}q_{i}q_{j},}
1398:
E. Ferapontov, A.P. Fordy, J. Geom. Phys., 21 (1997), p. 169
139:{\displaystyle u_{t}u_{xt}-u_{x}u_{tt}+u_{xx}+1=0\qquad (1)}
237:
are independent. The resulting system may always be put in
1044:{\displaystyle \lambda ^{N+1}=\prod _{i=0}^{N}(p-q_{i}),}
1407:
E.V Ferapontov, A.P Fordy, Physica D 108 (1997) 350-364
1340:
1313:
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1151:
1090:
1060:
975:
847:
813:
786:
759:
739:
719:
699:
666:
470:
335:
305:
278:
247:
216:
196:
176:
48:
1378:
Handbook of
Nonlinear Partial Differential Equations
1361:
satisfy the N-dimensional
Gibbons–Tsarev equations.
1353:
1326:
1296:
1203:
1170:
1130:
1073:
1043:
950:
826:
799:
772:
745:
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705:
678:
649:
455:
318:
291:
260:
229:
202:
182:
138:
693:from a fixed domain D, normally the complex half
1145:on the right hand side has N turning points,
241:form. Taking the characteristic speeds to be
8:
1376:Andrei D. Polyanin, Valentin F. Zaitsev,
1345:
1339:
1318:
1312:
1285:
1275:
1259:
1234:
1225:
1219:
1195:
1183:
1162:
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481:
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415:
402:
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361:
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310:
304:
299:, they are related to the zeroth moment
283:
277:
252:
246:
221:
215:
195:
175:
153:, as the condition that solutions of the
105:
89:
79:
63:
53:
47:
1432:Nonlinear partial differential equations
660:Both these equations hold for all pairs
1369:
1131:{\displaystyle \sum _{i=0}^{N}q_{i}=0.}
1204:{\displaystyle \lambda =\lambda _{i}}
166:Relationship to families of slit maps
149:The equation arises in the theory of
7:
838:describing the growth of each slit:
713:-plane, to a similar domain in the
1380:, second edition, p. 764 CRC PRESS
900:
885:
859:
851:
564:
545:
530:
502:
489:
474:
395:
380:
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339:
25:
151:dispersionless integrable systems
938:
634:
440:
126:
1035:
1016:
945:
939:
644:
635:
619:
592:
450:
441:
133:
127:
1:
37:partial differential equation
800:{\displaystyle \lambda _{i}}
773:{\displaystyle \lambda _{i}}
292:{\displaystyle \lambda _{i}}
753:and one variable end point
1448:
1054:where the real parameters
726:{\displaystyle \lambda }
1171:{\displaystyle p=p_{i}}
679:{\displaystyle i\neq j}
155:Benney moment equations
35:second order nonlinear
29:Gibbons–Tsarev equation
18:Gibbons-Tsarev equation
1355:
1328:
1298:
1205:
1172:
1132:
1111:
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1045:
1015:
952:
828:
801:
774:
747:
727:
707:
680:
651:
457:
320:
293:
268:and the corresponding
262:
231:
204:
184:
140:
1356:
1354:{\displaystyle A^{0}}
1329:
1327:{\displaystyle p_{i}}
1299:
1206:
1178:, with corresponding
1173:
1133:
1091:
1076:
1074:{\displaystyle q_{i}}
1046:
995:
953:
829:
827:{\displaystyle p_{i}}
802:
775:
748:
728:
708:
681:
652:
458:
321:
319:{\displaystyle A^{0}}
294:
263:
261:{\displaystyle p_{i}}
232:
230:{\displaystyle A^{n}}
205:
185:
141:
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1182:
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845:
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797:
780:; the preimage of
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270:Riemann invariants
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200:
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962:Analytic solution
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914:
873:
836:Loewner equations
746:{\displaystyle D}
706:{\displaystyle p}
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588:
559:
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239:Riemann invariant
203:{\displaystyle N}
183:{\displaystyle N}
16:(Redirected from
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691:conformal maps
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942:
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854:
841:
840:
839:
837:
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815:
792:
788:
765:
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720:
700:
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687:
673:
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638:
631:
623:
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605:
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596:
583:
579:
572:
568:
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534:
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518:
510:
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497:
493:
482:
478:
464:
447:
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429:
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416:
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388:
384:
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284:
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271:
253:
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123:
120:
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72:
67:
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60:
54:
50:
42:
41:
40:
38:
34:
30:
19:
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1403:
1394:
1385:
1377:
1372:
1306:
1140:
1053:
965:
688:
659:
169:
148:
28:
26:
1365:References
1143:polynomial
33:integrable
1257:∑
1253:∑
1193:λ
1186:λ
1093:∑
1081:satisfy:
1023:−
997:∏
978:λ
920:−
905:λ
901:∂
886:∂
879:−
864:λ
860:∂
852:∂
789:λ
762:λ
721:λ
671:≠
606:−
580:λ
565:∂
550:λ
546:∂
531:∂
507:λ
503:∂
494:λ
490:∂
475:∂
422:−
400:λ
396:∂
381:∂
374:−
359:λ
355:∂
340:∂
281:λ
73:−
1426:Category
1211:. With
272:to be
31:is an
1334:and
1307:the
1264:>
1141:The
326:by:
27:The
807:is
1428::
1126:0.
686:.
162:.
1347:0
1343:A
1320:i
1316:p
1292:,
1287:j
1283:q
1277:i
1273:q
1267:j
1261:i
1247:1
1244:+
1241:N
1237:1
1232:=
1227:0
1223:A
1197:i
1189:=
1164:i
1160:p
1156:=
1153:p
1123:=
1118:i
1114:q
1108:N
1103:0
1100:=
1097:i
1067:i
1063:q
1039:,
1036:)
1031:i
1027:q
1020:p
1017:(
1012:N
1007:0
1004:=
1001:i
993:=
988:1
985:+
982:N
946:)
943:3
940:(
936:.
928:i
924:p
917:p
909:j
894:0
890:A
876:=
868:i
855:p
820:i
816:p
793:i
766:i
741:D
701:p
674:j
668:i
645:)
642:b
639:2
636:(
632:.
624:2
620:)
614:j
610:p
601:i
597:p
593:(
584:j
573:0
569:A
554:i
539:0
535:A
522:2
519:=
511:j
498:i
483:0
479:A
451:)
448:a
445:2
442:(
438:,
430:j
426:p
417:i
413:p
404:j
389:0
385:A
371:=
363:j
348:i
344:p
312:0
308:A
285:i
254:i
250:p
223:n
219:A
198:N
178:N
134:)
131:1
128:(
124:0
121:=
118:1
115:+
110:x
107:x
103:u
99:+
94:t
91:t
87:u
81:x
77:u
68:t
65:x
61:u
55:t
51:u
20:)
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