1657:
449:
2312:
309:
1504:
1839:
1746:
707:
617:
899:
140:
1509:
315:
790:
1300:
1423:
1211:
1496:
498:, 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.
1148:
208:
1056:
941:
472:
45:
1010:
977:
219:
1652:{\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}}}
1361:
1334:
1083:
2113:
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).
1879:
1859:
492:
1751:
1665:
2009:
Calderbank, David M. J.; Kruglikov, Boris (2021). "Integrability via geometry: dispersionless differential equations in three and four dimensions".
1918:
2328:
1923:
495:
41:
2306:
628:
516:
1903:
1217:
796:
2054:
Kruglikov, Boris; Morozov, Oleg (2015). "Integrable
Dispersionless PDEs in 4D, Their Symmetry Pseudogroups and Deformations".
1928:
1306:
1233:
23:(PDE) arise in various problems of mathematics and physics and have been intensively studied in recent literature (see e.g.
20:
1913:
54:
444:{\displaystyle L_{2}=\partial _{t}+(\lambda ^{2}+u)\partial _{x}+(-\lambda u_{x}+u_{y})\partial _{\lambda },\qquad (3b)}
2333:
2164:
Konopelchenko, B. G. (2007). "Quasiclassical generalized
Weierstrass representation and dispersionless DS equation".
718:
27:
below). They typically arise when considering slowly modulated long waves of an integrable dispersive PDE system.
1242:
1369:
1220:. Such 'reductions', expressing the moments in terms of finitely many dependent variables, are described by the
1434:
1221:
1086:
1154:
1440:
1095:
2269:
2222:
2136:
2073:
1975:
2110:
Kodama Y., Gibbons J. "Integrability of the dispersionless KP hierarchy", Nonlinear World 1, (1990).
151:
1018:
2285:
2259:
2238:
2212:
2191:
2173:
2152:
2126:
2089:
2063:
2018:
1991:
1965:
1898:
2117:
Takasaki, Kanehisa; Takebe, Takashi (1995). "Integrable
Hierarchies and Dispersionless Limit".
304:{\displaystyle L_{1}=\partial _{y}+\lambda \partial _{x}-u_{x}\partial _{\lambda },\qquad (3a)}
2203:
Konopelchenko, B.G.; Moro, A. (2004). "Integrable
Equations in Nonlinear Geometrical Optics".
1908:
910:
457:
1889:
See for systems with contact Lax pairs, and e.g., and references therein for other systems.
982:
949:
2277:
2230:
2183:
2144:
2081:
2028:
1983:
2040:
1339:
1312:
1061:
2036:
2281:
1956:
Sergyeyev, A. (2018). "New integrable (3 + 1)-dimensional systems and contact geometry".
1834:{\displaystyle \partial _{\bar {z}}={\frac {1}{2}}(\partial _{x_{1}}+i\partial _{x_{2}})}
2273:
2226:
2140:
2077:
1979:
1216:
These may also be derived from considering slowly modulated wave train solutions of the
1864:
1844:
477:
2187:
2322:
2234:
2093:
1995:
2289:
2242:
2195:
2156:
507:
1741:{\displaystyle \partial _{z}={\frac {1}{2}}(\partial _{x_{1}}-i\partial _{x_{2}})}
2296:
Dunajski M. "Solitons, instantons and twistors", Oxford
University Press, 2010.
2032:
2250:
Dunajski, Maciej (2008). "An interpolating dispersionless integrable system".
2148:
2085:
1987:
1437:
is most commonly written as the following equation for a real-valued function
17:
44:(dKPE), also known (up to an inessential linear change of variables) as the
1336:-independent solutions of the dKP system. It is also obtainable from the
2131:
510:
moment hierarchy, each of which is a dispersionless integrable system:
2217:
2023:
1662:
where the following standard notation of complex analysis is used:
2264:
2178:
2068:
1970:
213:
of the following pair of 1-parameter families of vector fields
702:{\displaystyle \lambda =p+\sum _{n=0}^{\infty }A^{n}/p^{n+1},}
612:{\displaystyle A_{t_{2}}^{n}+A_{x}^{n+1}+nA^{n-1}A_{x}^{0}=0.}
2313:
1085:
are expressed in terms of just two functions, the classical
946:
and eliminating the other moments, as well as identifying
894:{\displaystyle p_{t_{3}}+p^{2}p_{x}+(pA^{0}+A^{1})_{x}=0,}
1305:
It is the dispersionless or quasiclassical limit of the
506:
The dispersionless KP system is closely related to the
1861:
here is an auxiliary function, defined uniquely from
712:
and the simplest two evolutions in the hierarchy are:
135:{\displaystyle (u_{t}+uu_{x})_{x}+u_{yy}=0,\qquad (1)}
1867:
1847:
1754:
1668:
1507:
1443:
1372:
1342:
1315:
1245:
1157:
1098:
1064:
1021:
985:
952:
913:
799:
721:
631:
519:
480:
460:
318:
222:
154:
57:
2252:Journal of Physics A: Mathematical and Theoretical
2166:Journal of Physics A: Mathematical and Theoretical
1885:Multidimensional integrable dispersionless systems
1873:
1853:
1833:
1740:
1651:
1490:
1417:
1355:
1328:
1294:
1205:
1142:
1077:
1050:
1004:
971:
935:
893:
784:
701:
611:
486:
466:
443:
303:
202:
134:
622:These arise as the consistency condition between
16:Dispersionless (or quasi-classical) limits of
785:{\displaystyle p_{t_{2}}+pp_{x}+A_{x}^{0}=0,}
8:
1295:{\displaystyle u_{t_{3}}=uu_{x}.\qquad (4)}
1418:{\displaystyle \lambda ^{2}=p^{2}+2A^{0}.}
1363:-flow of the Benney hierarchy on setting
2263:
2216:
2177:
2130:
2067:
2022:
1969:
1866:
1846:
1820:
1815:
1797:
1792:
1775:
1760:
1759:
1753:
1727:
1722:
1704:
1699:
1682:
1673:
1667:
1633:
1605:
1604:
1578:
1577:
1559:
1558:
1533:
1517:
1508:
1506:
1473:
1460:
1442:
1406:
1390:
1377:
1371:
1347:
1341:
1320:
1314:
1273:
1255:
1250:
1244:
1228:Dispersionless Korteweg–de Vries equation
1191:
1178:
1162:
1156:
1125:
1103:
1097:
1069:
1063:
1042:
1026:
1020:
996:
984:
963:
951:
924:
912:
876:
866:
853:
834:
824:
809:
804:
798:
767:
762:
749:
731:
726:
720:
684:
675:
669:
659:
648:
630:
597:
592:
576:
554:
549:
536:
529:
524:
518:
479:
474:is a spectral parameter. The dKPE is the
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419:
406:
393:
371:
352:
336:
323:
317:
279:
269:
256:
240:
227:
221:
175:
162:
153:
104:
91:
81:
65:
56:
1919:Dispersive partial differential equation
1429:Dispersionless Novikov–Veselov equation
494:-dispersionless limit of the celebrated
1945:
2011:Communications in Mathematical Physics
1206:{\displaystyle v_{y}+vv_{x}+h_{x}=0.}
1058:, so that the countably many moments
7:
1951:
1949:
1812:
1789:
1756:
1719:
1696:
1670:
1630:
1601:
1555:
1530:
1514:
1491:{\displaystyle v=v(x_{1},x_{2},t)}
660:
416:
368:
333:
276:
253:
237:
14:
1143:{\displaystyle h_{y}+(hv)_{x}=0,}
904:The dKP is recovered on setting
2309:at the dispersive equations wiki
2235:10.1111/j.0022-2526.2004.01536.x
2119:Reviews in Mathematical Physics
2056:Letters in Mathematical Physics
1958:Letters in Mathematical Physics
1924:Kadomtsev–Petviashvili equation
1282:
496:Kadomtsev–Petviashvili equation
428:
288:
190:
145:It arises from the commutation
122:
42:Kadomtsev–Petviashvili equation
24:
2329:Partial differential equations
2282:10.1088/1751-8113/41/31/315202
2205:Studies in Applied Mathematics
1904:Nonlinear Schrödinger equation
1828:
1785:
1765:
1735:
1692:
1610:
1589:
1583:
1571:
1564:
1548:
1539:
1485:
1453:
1289:
1283:
1218:nonlinear Schrödinger equation
1122:
1112:
873:
843:
438:
429:
412:
380:
364:
345:
298:
289:
197:
191:
181:
155:
129:
123:
88:
58:
46:Khokhlov–Zabolotskaya equation
21:partial differential equations
1:
1881:up to a holomorphic summand.
203:{\displaystyle =0.\qquad (2)}
1051:{\displaystyle A^{n}=hv^{n}}
2188:10.1088/1751-8113/40/46/F03
502:The Benney moment equations
2350:
2033:10.1007/s00220-020-03913-y
1929:Korteweg–de Vries equation
1307:Korteweg–de Vries equation
1234:Korteweg–de Vries equation
36:Dispersionless KP equation
2149:10.1142/S0129055X9500030X
2086:10.1007/s11005-015-0800-z
1988:10.1007/s11005-017-1013-4
1914:Davey–Stewartson equation
1435:Novikov-Veselov equation
936:{\displaystyle u=A^{0},}
467:{\displaystyle \lambda }
1222:Gibbons-Tsarev equation
1087:shallow water equations
1005:{\displaystyle t=t_{3}}
972:{\displaystyle y=t_{2}}
1875:
1855:
1835:
1742:
1653:
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1419:
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703:
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613:
488:
468:
445:
305:
204:
136:
1876:
1856:
1836:
1743:
1654:
1493:
1420:
1358:
1356:{\displaystyle t_{3}}
1331:
1329:{\displaystyle t_{2}}
1309:. It is satisfied by
1297:
1208:
1145:
1080:
1078:{\displaystyle A^{n}}
1053:
1007:
974:
938:
896:
787:
704:
644:
614:
489:
469:
446:
306:
205:
137:
1865:
1845:
1752:
1666:
1505:
1441:
1370:
1340:
1313:
1243:
1155:
1096:
1062:
1019:
983:
950:
911:
797:
719:
629:
517:
478:
458:
316:
220:
152:
55:
2274:2008JPhA...41E5202D
2227:2004nlin......3051K
2141:1995RvMaP...7..743T
2078:2015LMaPh.105.1703K
1980:2018LMaPh.108..359S
1433:The dispersionless
1232:The dispersionless
772:
602:
565:
541:
40:The dispersionless
2334:Integrable systems
2172:(46): F995–F1004.
1899:Integrable systems
1871:
1851:
1831:
1738:
1649:
1647:
1488:
1415:
1353:
1326:
1292:
1203:
1140:
1075:
1048:
1002:
969:
933:
891:
782:
758:
699:
609:
588:
545:
520:
484:
464:
441:
301:
200:
132:
2062:(12): 1703–1723.
1909:Nonlinear systems
1874:{\displaystyle v}
1854:{\displaystyle w}
1783:
1768:
1690:
1613:
1586:
1567:
1236:(dKdVE) reads as
487:{\displaystyle x}
2341:
2293:
2267:
2246:
2220:
2199:
2181:
2160:
2134:
2098:
2097:
2071:
2051:
2045:
2044:
2026:
2017:(3): 1811–1841.
2006:
2000:
1999:
1973:
1953:
1880:
1878:
1877:
1872:
1860:
1858:
1857:
1852:
1840:
1838:
1837:
1832:
1827:
1826:
1825:
1824:
1804:
1803:
1802:
1801:
1784:
1776:
1771:
1770:
1769:
1761:
1747:
1745:
1744:
1739:
1734:
1733:
1732:
1731:
1711:
1710:
1709:
1708:
1691:
1683:
1678:
1677:
1658:
1656:
1655:
1650:
1648:
1638:
1637:
1616:
1615:
1614:
1606:
1598:
1588:
1587:
1579:
1570:
1569:
1568:
1560:
1538:
1537:
1522:
1521:
1511:
1497:
1495:
1494:
1489:
1478:
1477:
1465:
1464:
1424:
1422:
1421:
1416:
1411:
1410:
1395:
1394:
1382:
1381:
1362:
1360:
1359:
1354:
1352:
1351:
1335:
1333:
1332:
1327:
1325:
1324:
1301:
1299:
1298:
1293:
1278:
1277:
1262:
1261:
1260:
1259:
1212:
1210:
1209:
1204:
1196:
1195:
1183:
1182:
1167:
1166:
1149:
1147:
1146:
1141:
1130:
1129:
1108:
1107:
1084:
1082:
1081:
1076:
1074:
1073:
1057:
1055:
1054:
1049:
1047:
1046:
1031:
1030:
1011:
1009:
1008:
1003:
1001:
1000:
978:
976:
975:
970:
968:
967:
942:
940:
939:
934:
929:
928:
900:
898:
897:
892:
881:
880:
871:
870:
858:
857:
839:
838:
829:
828:
816:
815:
814:
813:
791:
789:
788:
783:
771:
766:
754:
753:
738:
737:
736:
735:
708:
706:
705:
700:
695:
694:
679:
674:
673:
663:
658:
618:
616:
615:
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601:
596:
587:
586:
564:
553:
540:
535:
534:
533:
493:
491:
490:
485:
473:
471:
470:
465:
450:
448:
447:
442:
424:
423:
411:
410:
398:
397:
376:
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357:
356:
341:
340:
328:
327:
310:
308:
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284:
283:
274:
273:
261:
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245:
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232:
231:
209:
207:
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201:
180:
179:
167:
166:
141:
139:
138:
133:
112:
111:
96:
95:
86:
85:
70:
69:
2349:
2348:
2344:
2343:
2342:
2340:
2339:
2338:
2319:
2318:
2307:Ishimori_system
2303:
2249:
2202:
2163:
2116:
2107:
2102:
2101:
2053:
2052:
2048:
2008:
2007:
2003:
1955:
1954:
1947:
1942:
1937:
1895:
1887:
1863:
1862:
1843:
1842:
1841:. The function
1816:
1811:
1793:
1788:
1755:
1750:
1749:
1723:
1718:
1700:
1695:
1669:
1664:
1663:
1646:
1645:
1629:
1600:
1596:
1595:
1554:
1529:
1513:
1503:
1502:
1469:
1456:
1439:
1438:
1431:
1402:
1386:
1373:
1368:
1367:
1343:
1338:
1337:
1316:
1311:
1310:
1269:
1251:
1246:
1241:
1240:
1230:
1187:
1174:
1158:
1153:
1152:
1121:
1099:
1094:
1093:
1065:
1060:
1059:
1038:
1022:
1017:
1016:
992:
981:
980:
959:
948:
947:
920:
909:
908:
872:
862:
849:
830:
820:
805:
800:
795:
794:
745:
727:
722:
717:
716:
680:
665:
627:
626:
572:
525:
515:
514:
504:
476:
475:
456:
455:
415:
402:
389:
367:
348:
332:
319:
314:
313:
275:
265:
252:
236:
223:
218:
217:
171:
158:
150:
149:
100:
87:
77:
61:
53:
52:
48:, has the form
38:
33:
12:
11:
5:
2347:
2345:
2337:
2336:
2331:
2321:
2320:
2317:
2316:
2310:
2302:
2301:External links
2299:
2298:
2297:
2294:
2258:(31): 315202.
2247:
2211:(4): 325–352.
2200:
2161:
2132:hep-th/9405096
2125:(5): 743–808.
2114:
2111:
2106:
2103:
2100:
2099:
2046:
2001:
1964:(2): 359–376.
1944:
1943:
1941:
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2105:Bibliography
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2323:Categories
2024:1612.02753
1935:References
25:references
18:integrable
2265:0804.1234
2179:0709.4148
2094:119326497
2069:1410.7104
1996:119159629
1971:1401.2122
1940:Citations
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1893:See also
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31:Examples
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