1668:
460:
2323:
320:
1515:
1850:
1757:
718:
628:
910:
151:
1520:
326:
801:
1311:
1434:
1222:
1507:
509:, arising when considering long waves of that system. The dKPE, like many other (2+1)-dimensional integrable dispersionless systems, admits a (3+1)-dimensional generalization.
1159:
219:
1067:
952:
483:
56:
1021:
988:
230:
1663:{\displaystyle {\begin{aligned}&\partial _{t}v=\partial _{z}(vw)+\partial _{\bar {z}}(v{\bar {w}}),\\&\partial _{\bar {z}}w=-3\partial _{z}v,\end{aligned}}}
1372:
1345:
1094:
2124:
Zakharov V.E. "Dispersionless limit of integrable systems in 2+1 dimensions", Singular Limits of
Dispersive Waves, NATO ASI series, Volume 320, 165-174, (1994).
1890:
1870:
503:
1762:
1676:
2020:
Calderbank, David M. J.; Kruglikov, Boris (2021). "Integrability via geometry: dispersionless differential equations in three and four dimensions".
1929:
2339:
1934:
506:
52:
2317:
639:
527:
1914:
1228:
807:
2065:
Kruglikov, Boris; Morozov, Oleg (2015). "Integrable
Dispersionless PDEs in 4D, Their Symmetry Pseudogroups and Deformations".
1939:
1317:
1244:
34:(PDE) arise in various problems of mathematics and physics and have been intensively studied in recent literature (see e.g.
31:
1924:
65:
455:{\displaystyle L_{2}=\partial _{t}+(\lambda ^{2}+u)\partial _{x}+(-\lambda u_{x}+u_{y})\partial _{\lambda },\qquad (3b)}
2344:
2175:
Konopelchenko, B. G. (2007). "Quasiclassical generalized
Weierstrass representation and dispersionless DS equation".
729:
38:
below). They typically arise when considering slowly modulated long waves of an integrable dispersive PDE system.
1253:
1380:
1231:. Such 'reductions', expressing the moments in terms of finitely many dependent variables, are described by the
1445:
1232:
1097:
1165:
1451:
1106:
2280:
2233:
2147:
2084:
1986:
2121:
Kodama Y., Gibbons J. "Integrability of the dispersionless KP hierarchy", Nonlinear World 1, (1990).
162:
1029:
2296:
2270:
2249:
2223:
2202:
2184:
2163:
2137:
2100:
2074:
2029:
2002:
1976:
1909:
2128:
Takasaki, Kanehisa; Takebe, Takashi (1995). "Integrable
Hierarchies and Dispersionless Limit".
315:{\displaystyle L_{1}=\partial _{y}+\lambda \partial _{x}-u_{x}\partial _{\lambda },\qquad (3a)}
2214:
Konopelchenko, B.G.; Moro, A. (2004). "Integrable
Equations in Nonlinear Geometrical Optics".
1919:
921:
468:
1900:
See for systems with contact Lax pairs, and e.g., and references therein for other systems.
993:
960:
2288:
2241:
2194:
2155:
2092:
2039:
1994:
2051:
1350:
1323:
1072:
2047:
2292:
1967:
Sergyeyev, A. (2018). "New integrable (3 + 1)-dimensional systems and contact geometry".
1845:{\displaystyle \partial _{\bar {z}}={\frac {1}{2}}(\partial _{x_{1}}+i\partial _{x_{2}})}
2284:
2237:
2151:
2088:
1990:
1227:
These may also be derived from considering slowly modulated wave train solutions of the
17:
1875:
1855:
488:
2198:
2333:
2245:
2104:
2006:
2300:
2253:
2206:
2167:
518:
1752:{\displaystyle \partial _{z}={\frac {1}{2}}(\partial _{x_{1}}-i\partial _{x_{2}})}
2307:
Dunajski M. "Solitons, instantons and twistors", Oxford
University Press, 2010.
2043:
2261:
Dunajski, Maciej (2008). "An interpolating dispersionless integrable system".
2159:
2096:
1998:
1448:
is most commonly written as the following equation for a real-valued function
28:
55:(dKPE), also known (up to an inessential linear change of variables) as the
1347:-independent solutions of the dKP system. It is also obtainable from the
2142:
521:
moment hierarchy, each of which is a dispersionless integrable system:
2228:
2034:
1673:
where the following standard notation of complex analysis is used:
2275:
2189:
2079:
1981:
224:
of the following pair of 1-parameter families of vector fields
713:{\displaystyle \lambda =p+\sum _{n=0}^{\infty }A^{n}/p^{n+1},}
623:{\displaystyle A_{t_{2}}^{n}+A_{x}^{n+1}+nA^{n-1}A_{x}^{0}=0.}
2324:
1096:
are expressed in terms of just two functions, the classical
957:
and eliminating the other moments, as well as identifying
905:{\displaystyle p_{t_{3}}+p^{2}p_{x}+(pA^{0}+A^{1})_{x}=0,}
1316:
It is the dispersionless or quasiclassical limit of the
517:
The dispersionless KP system is closely related to the
1872:
here is an auxiliary function, defined uniquely from
723:
and the simplest two evolutions in the hierarchy are:
146:{\displaystyle (u_{t}+uu_{x})_{x}+u_{yy}=0,\qquad (1)}
1878:
1858:
1765:
1679:
1518:
1454:
1383:
1353:
1326:
1256:
1168:
1109:
1075:
1032:
996:
963:
924:
810:
732:
642:
530:
491:
471:
329:
233:
165:
68:
2263:Journal of Physics A: Mathematical and Theoretical
2177:Journal of Physics A: Mathematical and Theoretical
1896:Multidimensional integrable dispersionless systems
1884:
1864:
1844:
1751:
1662:
1501:
1428:
1366:
1339:
1305:
1216:
1153:
1088:
1061:
1015:
982:
946:
904:
795:
712:
622:
497:
477:
454:
314:
213:
145:
633:These arise as the consistency condition between
27:Dispersionless (or quasi-classical) limits of
796:{\displaystyle p_{t_{2}}+pp_{x}+A_{x}^{0}=0,}
8:
1306:{\displaystyle u_{t_{3}}=uu_{x}.\qquad (4)}
1429:{\displaystyle \lambda ^{2}=p^{2}+2A^{0}.}
1374:-flow of the Benney hierarchy on setting
2274:
2227:
2188:
2141:
2078:
2033:
1980:
1877:
1857:
1831:
1826:
1808:
1803:
1786:
1771:
1770:
1764:
1738:
1733:
1715:
1710:
1693:
1684:
1678:
1644:
1616:
1615:
1589:
1588:
1570:
1569:
1544:
1528:
1519:
1517:
1484:
1471:
1453:
1417:
1401:
1388:
1382:
1358:
1352:
1331:
1325:
1284:
1266:
1261:
1255:
1239:Dispersionless Korteweg–de Vries equation
1202:
1189:
1173:
1167:
1136:
1114:
1108:
1080:
1074:
1053:
1037:
1031:
1007:
995:
974:
962:
935:
923:
887:
877:
864:
845:
835:
820:
815:
809:
778:
773:
760:
742:
737:
731:
695:
686:
680:
670:
659:
641:
608:
603:
587:
565:
560:
547:
540:
535:
529:
490:
485:is a spectral parameter. The dKPE is the
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430:
417:
404:
382:
363:
347:
334:
328:
290:
280:
267:
251:
238:
232:
186:
173:
164:
115:
102:
92:
76:
67:
1930:Dispersive partial differential equation
1440:Dispersionless Novikov–Veselov equation
505:-dispersionless limit of the celebrated
1956:
2022:Communications in Mathematical Physics
1217:{\displaystyle v_{y}+vv_{x}+h_{x}=0.}
1069:, so that the countably many moments
7:
1962:
1960:
1823:
1800:
1767:
1730:
1707:
1681:
1641:
1612:
1566:
1541:
1525:
1502:{\displaystyle v=v(x_{1},x_{2},t)}
671:
427:
379:
344:
287:
264:
248:
25:
1154:{\displaystyle h_{y}+(hv)_{x}=0,}
915:The dKP is recovered on setting
2320:at the dispersive equations wiki
2246:10.1111/j.0022-2526.2004.01536.x
2130:Reviews in Mathematical Physics
2067:Letters in Mathematical Physics
1969:Letters in Mathematical Physics
1935:Kadomtsev–Petviashvili equation
1293:
507:Kadomtsev–Petviashvili equation
439:
299:
201:
156:It arises from the commutation
133:
53:Kadomtsev–Petviashvili equation
35:
2340:Partial differential equations
2293:10.1088/1751-8113/41/31/315202
2216:Studies in Applied Mathematics
1915:Nonlinear Schrödinger equation
1839:
1796:
1776:
1746:
1703:
1621:
1600:
1594:
1582:
1575:
1559:
1550:
1496:
1464:
1300:
1294:
1229:nonlinear Schrödinger equation
1133:
1123:
884:
854:
449:
440:
423:
391:
375:
356:
309:
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208:
202:
192:
166:
140:
134:
99:
69:
57:Khokhlov–Zabolotskaya equation
32:partial differential equations
1:
1892:up to a holomorphic summand.
214:{\displaystyle =0.\qquad (2)}
1062:{\displaystyle A^{n}=hv^{n}}
2199:10.1088/1751-8113/40/46/F03
513:The Benney moment equations
2361:
2044:10.1007/s00220-020-03913-y
1940:Korteweg–de Vries equation
1318:Korteweg–de Vries equation
1245:Korteweg–de Vries equation
47:Dispersionless KP equation
2160:10.1142/S0129055X9500030X
2097:10.1007/s11005-015-0800-z
1999:10.1007/s11005-017-1013-4
1925:Davey–Stewartson equation
1446:Novikov-Veselov equation
947:{\displaystyle u=A^{0},}
478:{\displaystyle \lambda }
18:Dispersionless equations
1233:Gibbons-Tsarev equation
1098:shallow water equations
1016:{\displaystyle t=t_{3}}
983:{\displaystyle y=t_{2}}
1886:
1866:
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215:
147:
1887:
1867:
1847:
1754:
1665:
1504:
1431:
1369:
1367:{\displaystyle t_{3}}
1342:
1340:{\displaystyle t_{2}}
1320:. It is satisfied by
1308:
1219:
1156:
1091:
1089:{\displaystyle A^{n}}
1064:
1018:
985:
949:
907:
798:
715:
655:
625:
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457:
317:
216:
148:
1876:
1856:
1763:
1677:
1516:
1452:
1381:
1351:
1324:
1254:
1166:
1107:
1073:
1030:
994:
961:
922:
808:
730:
640:
528:
489:
469:
327:
231:
163:
66:
2285:2008JPhA...41E5202D
2238:2004nlin......3051K
2152:1995RvMaP...7..743T
2089:2015LMaPh.105.1703K
1991:2018LMaPh.108..359S
1444:The dispersionless
1243:The dispersionless
783:
613:
576:
552:
51:The dispersionless
2345:Integrable systems
2183:(46): F995–F1004.
1910:Integrable systems
1882:
1862:
1842:
1749:
1660:
1658:
1499:
1426:
1364:
1337:
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902:
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531:
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475:
452:
312:
211:
143:
2073:(12): 1703–1723.
1920:Nonlinear systems
1885:{\displaystyle v}
1865:{\displaystyle w}
1794:
1779:
1701:
1624:
1597:
1578:
1247:(dKdVE) reads as
498:{\displaystyle x}
16:(Redirected from
2352:
2304:
2278:
2257:
2231:
2210:
2192:
2171:
2145:
2109:
2108:
2082:
2062:
2056:
2055:
2037:
2028:(3): 1811–1841.
2017:
2011:
2010:
1984:
1964:
1891:
1889:
1888:
1883:
1871:
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1868:
1863:
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1702:
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1688:
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1659:
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1627:
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1609:
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1522:
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1422:
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1406:
1405:
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1365:
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1362:
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1344:
1343:
1338:
1336:
1335:
1312:
1310:
1309:
1304:
1289:
1288:
1273:
1272:
1271:
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1223:
1221:
1220:
1215:
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1206:
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1160:
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1141:
1140:
1119:
1118:
1095:
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1092:
1087:
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1066:
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1060:
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1057:
1042:
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1019:
1014:
1012:
1011:
989:
987:
986:
981:
979:
978:
953:
951:
950:
945:
940:
939:
911:
909:
908:
903:
892:
891:
882:
881:
869:
868:
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849:
840:
839:
827:
826:
825:
824:
802:
800:
799:
794:
782:
777:
765:
764:
749:
748:
747:
746:
719:
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690:
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684:
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629:
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621:
612:
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551:
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545:
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501:
496:
484:
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387:
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97:
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81:
80:
21:
2360:
2359:
2355:
2354:
2353:
2351:
2350:
2349:
2330:
2329:
2318:Ishimori_system
2314:
2260:
2213:
2174:
2127:
2118:
2113:
2112:
2064:
2063:
2059:
2019:
2018:
2014:
1966:
1965:
1958:
1953:
1948:
1906:
1898:
1874:
1873:
1854:
1853:
1852:. The function
1827:
1822:
1804:
1799:
1766:
1761:
1760:
1734:
1729:
1711:
1706:
1680:
1675:
1674:
1657:
1656:
1640:
1611:
1607:
1606:
1565:
1540:
1524:
1514:
1513:
1480:
1467:
1450:
1449:
1442:
1413:
1397:
1384:
1379:
1378:
1354:
1349:
1348:
1327:
1322:
1321:
1280:
1262:
1257:
1252:
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1241:
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1132:
1110:
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1104:
1076:
1071:
1070:
1049:
1033:
1028:
1027:
1003:
992:
991:
970:
959:
958:
931:
920:
919:
883:
873:
860:
841:
831:
816:
811:
806:
805:
756:
738:
733:
728:
727:
691:
676:
638:
637:
583:
536:
526:
525:
515:
487:
486:
467:
466:
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286:
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263:
247:
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229:
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182:
169:
161:
160:
111:
98:
88:
72:
64:
63:
59:, has the form
49:
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23:
22:
15:
12:
11:
5:
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2356:
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2342:
2332:
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2313:
2312:External links
2310:
2309:
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2305:
2269:(31): 315202.
2258:
2222:(4): 325–352.
2211:
2172:
2143:hep-th/9405096
2136:(5): 743–808.
2125:
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2110:
2057:
2012:
1975:(2): 359–376.
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2334:Categories
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1946:References
36:references
29:integrable
2276:0804.1234
2190:0709.4148
2105:119326497
2080:1410.7104
2007:119159629
1982:1401.2122
1951:Citations
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