968:
651:
1515:
963:{\displaystyle {\begin{aligned}{\frac {\partial R}{\partial t}}&=-\alpha (2\nabla \Theta \cdot \nabla R+R\nabla ^{2}\Theta )+(1-R^{2})R,\\R{\frac {\partial \Theta }{\partial t}}&=\alpha +(2\nabla \Theta \cdot \nabla R+R\nabla ^{2}\Theta )-\beta R^{3}.\end{aligned}}}
2303:
2519:
Hocking, L. M., & Stewartson, K. (1972). On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance. Proceedings of the Royal
Society of London. A. Mathematical and Physical Sciences, 326(1566),
1142:
1916:
423:
2118:
2035:
1364:
1789:
1673:
265:
2362:
2159:
2167:
1039:
1320:
2431:
656:
1579:
564:
643:
1251:
1186:
1047:
1800:
454:
474:
307:
600:
91:
64:
2510:
Stewartson, K., & Stuart, J. T. (1971). A non-linear instability theory for a wave system in plane
Poiseuille flow. Journal of Fluid Mechanics, 48(3), 529-545.
1219:
1343:
497:
296:
174:
151:
131:
111:
2048:
1922:
1510:{\displaystyle A(\mathbf {x} ,t)={\sqrt {1-k^{2}}}e^{i\mathbf {k} \cdot \mathbf {x} -\omega t},\quad \omega =\beta +(\alpha -\beta )k^{2},\quad |k|<1.}
39:
from a stable to an unstable state of a system. At the onset of finite wavelength bifurcation, the system becomes unstable for a critical wavenumber
1692:
1588:
66:
which is non-zero. In the neighbourhood of this bifurcation, the evolution of disturbances is characterised by the particular
Fourier mode for
185:
2501:
Segel, L. A. (1969). Distant side-walls cause slow amplitude modulation of cellular convection. Journal of Fluid
Mechanics, 38(1), 203-224.
1682:
Hocking–Stewartson pulse refers to a quasi-steady, 1D solution of the complex
Ginzburg–Landau equation, obtained by Leslie M. Hocking and
2492:
Newell, A. C., & Whitehead, J. A. (1969). Finite bandwidth, finite amplitude convection. Journal of Fluid
Mechanics, 38(2), 279-303.
2311:
2123:
2298:{\displaystyle {\frac {\partial B}{\partial t}}+\mathbf {v} \cdot \nabla B=(1+i\alpha )\nabla ^{2}B+\lambda B-(1+i\beta )B|B|^{2}}
986:
1259:
2474:
Cross, M. C., & Hohenberg, P. C. (1993). Pattern formation outside of equilibrium. Reviews of modern physics, 65(3), 851.
2483:
Cross, M., & Greenside, H. (2009). Pattern formation and dynamics in nonequilibrium systems. Cambridge
University Press.
2456:
2441:
2367:
1582:
2451:
504:
2446:
1523:
513:
500:
17:
605:
2535:
1224:
1345:
is an arbitrary real constant. Similar solutions can be constructed numerically in a finite domain.
1137:{\displaystyle A(\mathbf {x} )={\sqrt {1-k^{2}}}e^{i\mathbf {k} \cdot \mathbf {x} },\quad |k|<1.}
2545:
1150:
1911:{\displaystyle \lambda ^{2}(M^{2}+2\alpha -1)=1,\quad \lambda ^{2}(\alpha -\alpha M^{2}+2M)=\nu ,}
2540:
433:
36:
418:{\displaystyle {\frac {\partial A}{\partial t}}=(1+i\alpha )\nabla ^{2}A+A-(1+i\beta )A|A|^{2}}
439:
1683:
459:
429:
569:
69:
42:
28:
1198:
2113:{\displaystyle A=e^{i\mathbf {k} \cdot \mathbf {x} -\omega t}B({\boldsymbol {\xi }},t)}
1328:
1195:
Once again, let us look for steady solutions, but with an absorbing boundary condition
482:
281:
271:
159:
136:
116:
96:
2529:
2030:{\displaystyle 2-M^{2}-3\alpha M=-L^{2},\quad 2\alpha +3M-\alpha M^{2}=-\beta L^{2}.}
1147:
This solution is known to become unstable due to
Eckhaus instability for wavenumbers
499:
is independent of the spatial coordinates, the
Ginzburg–Landau equation reduces to
35:, describes the nonlinear evolution of small disturbances near a finite wavelength
153:. The unstable modes can either be non-oscillatory (stationary) or oscillatory.
32:
275:
1784:{\displaystyle A(x,t)=\lambda Le^{i\nu t}(\mathrm {sech} \lambda x)^{1+iM}}
1668:{\displaystyle k^{2}>(1+\alpha \beta )/(2\beta ^{2}+\alpha \beta +3).}
260:{\displaystyle {\frac {\partial A}{\partial t}}=\nabla ^{2}A+A-A|A|^{2}}
2357:{\displaystyle \mathbf {v} =\mathbf {u} +(1+i\alpha )\mathbf {k} }
1041:
in the real equation without the time derivative term, we obtain
2154:{\displaystyle {\boldsymbol {\xi }}=\mathbf {x} +\mathbf {u} t}
133:). The Ginzburg–Landau equation is the governing equation for
1794:
where the four real constants in the above solution satisfy
2045:
The coherent structure solutions are obtained by assuming
1034:{\displaystyle A=f(k)e^{i\mathbf {k} \cdot \mathbf {x} }}
1315:{\displaystyle A(x)=e^{ia}\tanh {\frac {x}{\sqrt {2}}},}
2426:{\displaystyle \lambda =1+i\omega -(1+i\alpha )k^{2}.}
1349:
Some solutions of the complex
Ginzburg–Landau equation
2370:
2314:
2170:
2126:
2051:
1925:
1803:
1695:
1591:
1526:
1367:
1331:
1262:
1227:
1201:
1153:
1050:
989:
654:
608:
572:
516:
485:
462:
442:
310:
284:
188:
162:
139:
119:
99:
72:
45:
974:
Some solutions of the real Ginzburg–Landau equation
2425:
2356:
2297:
2153:
2112:
2029:
1910:
1783:
1667:
1573:
1509:
1337:
1314:
1245:
1213:
1180:
1136:
1033:
962:
645:is the phase, one obtains the following equations
637:
594:
558:
491:
468:
448:
417:
290:
259:
168:
145:
125:
105:
85:
58:
1191:Steady solution with absorbing boundary condition
1221:at some location. In a semi-infinite, 1D domain
1574:{\displaystyle d\omega /dk=2(\alpha -\beta )k.}
559:{\displaystyle A(\mathbf {x} ,t)=Re^{i\Theta }}
8:
479:When the problem is homogeneous, i.e., when
1581:The above solution becomes unstable due to
1520:The group velocity of the wave is given by
2414:
2369:
2349:
2323:
2315:
2313:
2289:
2284:
2275:
2233:
2194:
2171:
2169:
2143:
2135:
2127:
2125:
2096:
2074:
2066:
2062:
2050:
2018:
1999:
1964:
1936:
1924:
1881:
1859:
1821:
1808:
1802:
1766:
1742:
1727:
1694:
1638:
1623:
1596:
1590:
1533:
1525:
1496:
1488:
1478:
1427:
1419:
1415:
1403:
1391:
1374:
1366:
1330:
1297:
1282:
1261:
1226:
1200:
1170:
1158:
1152:
1123:
1115:
1104:
1096:
1092:
1080:
1068:
1057:
1049:
1024:
1016:
1012:
988:
947:
925:
882:
854:
817:
795:
767:
721:
693:
659:
655:
653:
638:{\displaystyle \Theta =\mathrm {arg} (A)}
615:
607:
587:
579:
571:
547:
523:
515:
507:results in the Ginzburg–Landau equation.
484:
461:
441:
409:
404:
395:
356:
311:
309:
283:
251:
246:
237:
216:
189:
187:
161:
138:
118:
98:
77:
71:
50:
44:
1358:The traveling wave solution is given by
2467:
2128:
2097:
278:in 1969. For oscillatory bifurcation,
16:For the superconductivity theory, see
7:
2230:
2202:
2182:
2174:
1752:
1749:
1746:
1743:
1686:in 1972. The solution is given by
1246:{\displaystyle 0\leq x<\infty }
1240:
931:
922:
909:
903:
900:
875:
872:
851:
828:
823:
820:
773:
764:
751:
745:
742:
714:
711:
690:
670:
662:
622:
619:
616:
609:
551:
353:
322:
314:
213:
200:
192:
14:
156:For non-oscillatory bifurcation,
113:(more precisely the real part of
2350:
2324:
2316:
2195:
2144:
2136:
2075:
2067:
1428:
1420:
1375:
1105:
1097:
1058:
1025:
1017:
524:
300:complex Ginzburg–Landau equation
1973:
1854:
1487:
1446:
1114:
2407:
2392:
2346:
2331:
2285:
2276:
2269:
2254:
2226:
2211:
2107:
2093:
1896:
1865:
1842:
1814:
1763:
1739:
1711:
1699:
1659:
1628:
1620:
1605:
1562:
1550:
1497:
1489:
1471:
1459:
1385:
1371:
1272:
1266:
1124:
1116:
1062:
1054:
1005:
999:
934:
894:
888:
879:
869:
847:
801:
782:
776:
736:
727:
718:
708:
686:
632:
626:
588:
580:
534:
520:
405:
396:
389:
374:
349:
334:
247:
238:
93:with slowly varying amplitude
1:
1181:{\displaystyle k^{2}>1/3.}
274:and John A. Whitehead and by
178:real Ginzburg–Landau equation
2041:Coherent structure solutions
1253:, the solution is given by
428:which was first derived by
270:which was first derived by
2562:
15:
2457:Gross–Pitaevskii equation
2442:Davey–Stewartson equation
1583:Benjamin–Feir instability
2452:Swift–Hohenberg equation
1678:Hocking–Stewartson pulse
505:Swift–Hohenberg equation
25:Ginzburg–Landau equation
449:{\displaystyle \alpha }
2447:Stuart–Landau equation
2427:
2358:
2299:
2155:
2114:
2031:
1912:
1785:
1669:
1575:
1511:
1339:
1316:
1247:
1215:
1182:
1138:
1035:
979:Steady plane-wave type
964:
639:
596:
560:
501:Stuart–Landau equation
493:
470:
469:{\displaystyle \beta }
450:
419:
292:
261:
170:
147:
127:
107:
87:
60:
18:Ginzburg–Landau theory
2428:
2359:
2300:
2156:
2115:
2032:
1913:
1786:
1670:
1576:
1512:
1340:
1317:
1248:
1216:
1183:
1139:
1036:
965:
640:
602:is the amplitude and
597:
595:{\displaystyle R=|A|}
561:
494:
471:
451:
420:
293:
262:
171:
148:
128:
108:
88:
86:{\displaystyle k_{c}}
61:
59:{\displaystyle k_{c}}
2368:
2312:
2168:
2124:
2049:
1923:
1801:
1693:
1589:
1524:
1365:
1329:
1260:
1225:
1199:
1151:
1048:
987:
652:
606:
570:
514:
483:
476:are real constants.
460:
440:
308:
282:
186:
160:
137:
117:
97:
70:
43:
1214:{\displaystyle A=0}
2423:
2354:
2295:
2151:
2110:
2027:
1908:
1781:
1665:
1571:
1507:
1335:
1312:
1243:
1211:
1178:
1134:
1031:
960:
958:
635:
592:
556:
489:
466:
446:
434:John Trevor Stuart
415:
288:
257:
166:
143:
123:
103:
83:
56:
2189:
1409:
1338:{\displaystyle a}
1307:
1306:
1086:
983:If we substitute
835:
677:
492:{\displaystyle A}
329:
291:{\displaystyle A}
207:
169:{\displaystyle A}
146:{\displaystyle A}
126:{\displaystyle A}
106:{\displaystyle A}
2553:
2521:
2517:
2511:
2508:
2502:
2499:
2493:
2490:
2484:
2481:
2475:
2472:
2432:
2430:
2429:
2424:
2419:
2418:
2363:
2361:
2360:
2355:
2353:
2327:
2319:
2304:
2302:
2301:
2296:
2294:
2293:
2288:
2279:
2238:
2237:
2198:
2190:
2188:
2180:
2172:
2161:. This leads to
2160:
2158:
2157:
2152:
2147:
2139:
2131:
2119:
2117:
2116:
2111:
2100:
2089:
2088:
2078:
2070:
2036:
2034:
2033:
2028:
2023:
2022:
2004:
2003:
1969:
1968:
1941:
1940:
1917:
1915:
1914:
1909:
1886:
1885:
1864:
1863:
1826:
1825:
1813:
1812:
1790:
1788:
1787:
1782:
1780:
1779:
1755:
1738:
1737:
1684:Keith Stewartson
1674:
1672:
1671:
1666:
1643:
1642:
1627:
1601:
1600:
1585:for wavenumbers
1580:
1578:
1577:
1572:
1537:
1516:
1514:
1513:
1508:
1500:
1492:
1483:
1482:
1442:
1441:
1431:
1423:
1410:
1408:
1407:
1392:
1378:
1344:
1342:
1341:
1336:
1321:
1319:
1318:
1313:
1308:
1302:
1298:
1290:
1289:
1252:
1250:
1249:
1244:
1220:
1218:
1217:
1212:
1187:
1185:
1184:
1179:
1174:
1163:
1162:
1143:
1141:
1140:
1135:
1127:
1119:
1110:
1109:
1108:
1100:
1087:
1085:
1084:
1069:
1061:
1040:
1038:
1037:
1032:
1030:
1029:
1028:
1020:
969:
967:
966:
961:
959:
952:
951:
930:
929:
887:
886:
859:
858:
836:
834:
826:
818:
800:
799:
772:
771:
726:
725:
698:
697:
678:
676:
668:
660:
644:
642:
641:
636:
625:
601:
599:
598:
593:
591:
583:
565:
563:
562:
557:
555:
554:
527:
498:
496:
495:
490:
475:
473:
472:
467:
455:
453:
452:
447:
430:Keith Stewartson
424:
422:
421:
416:
414:
413:
408:
399:
361:
360:
330:
328:
320:
312:
297:
295:
294:
289:
266:
264:
263:
258:
256:
255:
250:
241:
221:
220:
208:
206:
198:
190:
175:
173:
172:
167:
152:
150:
149:
144:
132:
130:
129:
124:
112:
110:
109:
104:
92:
90:
89:
84:
82:
81:
65:
63:
62:
57:
55:
54:
2561:
2560:
2556:
2555:
2554:
2552:
2551:
2550:
2526:
2525:
2524:
2518:
2514:
2509:
2505:
2500:
2496:
2491:
2487:
2482:
2478:
2473:
2469:
2465:
2438:
2410:
2366:
2365:
2310:
2309:
2283:
2229:
2181:
2173:
2166:
2165:
2122:
2121:
2058:
2047:
2046:
2043:
2014:
1995:
1960:
1932:
1921:
1920:
1877:
1855:
1817:
1804:
1799:
1798:
1762:
1723:
1691:
1690:
1680:
1634:
1592:
1587:
1586:
1522:
1521:
1474:
1411:
1399:
1363:
1362:
1356:
1351:
1327:
1326:
1278:
1258:
1257:
1223:
1222:
1197:
1196:
1193:
1154:
1149:
1148:
1088:
1076:
1046:
1045:
1008:
985:
984:
981:
976:
957:
956:
943:
921:
878:
850:
837:
827:
819:
811:
810:
791:
763:
717:
689:
679:
669:
661:
650:
649:
604:
603:
568:
567:
543:
512:
511:
481:
480:
458:
457:
438:
437:
403:
352:
321:
313:
306:
305:
280:
279:
245:
212:
199:
191:
184:
183:
158:
157:
135:
134:
115:
114:
95:
94:
73:
68:
67:
46:
41:
40:
29:Vitaly Ginzburg
21:
12:
11:
5:
2559:
2557:
2549:
2548:
2543:
2538:
2536:Fluid dynamics
2528:
2527:
2523:
2522:
2512:
2503:
2494:
2485:
2476:
2466:
2464:
2461:
2460:
2459:
2454:
2449:
2444:
2437:
2434:
2422:
2417:
2413:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2352:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2326:
2322:
2318:
2306:
2305:
2292:
2287:
2282:
2278:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2236:
2232:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2197:
2193:
2187:
2184:
2179:
2176:
2150:
2146:
2142:
2138:
2134:
2130:
2109:
2106:
2103:
2099:
2095:
2092:
2087:
2084:
2081:
2077:
2073:
2069:
2065:
2061:
2057:
2054:
2042:
2039:
2038:
2037:
2026:
2021:
2017:
2013:
2010:
2007:
2002:
1998:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1972:
1967:
1963:
1959:
1956:
1953:
1950:
1947:
1944:
1939:
1935:
1931:
1928:
1918:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1884:
1880:
1876:
1873:
1870:
1867:
1862:
1858:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1824:
1820:
1816:
1811:
1807:
1792:
1791:
1778:
1775:
1772:
1769:
1765:
1761:
1758:
1754:
1751:
1748:
1745:
1741:
1736:
1733:
1730:
1726:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1679:
1676:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1641:
1637:
1633:
1630:
1626:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1599:
1595:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1536:
1532:
1529:
1518:
1517:
1506:
1503:
1499:
1495:
1491:
1486:
1481:
1477:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1445:
1440:
1437:
1434:
1430:
1426:
1422:
1418:
1414:
1406:
1402:
1398:
1395:
1390:
1387:
1384:
1381:
1377:
1373:
1370:
1355:
1354:Traveling wave
1352:
1350:
1347:
1334:
1323:
1322:
1311:
1305:
1301:
1296:
1293:
1288:
1285:
1281:
1277:
1274:
1271:
1268:
1265:
1242:
1239:
1236:
1233:
1230:
1210:
1207:
1204:
1192:
1189:
1177:
1173:
1169:
1166:
1161:
1157:
1145:
1144:
1133:
1130:
1126:
1122:
1118:
1113:
1107:
1103:
1099:
1095:
1091:
1083:
1079:
1075:
1072:
1067:
1064:
1060:
1056:
1053:
1027:
1023:
1019:
1015:
1011:
1007:
1004:
1001:
998:
995:
992:
980:
977:
975:
972:
971:
970:
955:
950:
946:
942:
939:
936:
933:
928:
924:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
885:
881:
877:
874:
871:
868:
865:
862:
857:
853:
849:
846:
843:
840:
838:
833:
830:
825:
822:
816:
813:
812:
809:
806:
803:
798:
794:
790:
787:
784:
781:
778:
775:
770:
766:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
732:
729:
724:
720:
716:
713:
710:
707:
704:
701:
696:
692:
688:
685:
682:
680:
675:
672:
667:
664:
658:
657:
634:
631:
628:
624:
621:
618:
614:
611:
590:
586:
582:
578:
575:
553:
550:
546:
542:
539:
536:
533:
530:
526:
522:
519:
488:
465:
445:
436:in 1971. Here
426:
425:
412:
407:
402:
398:
394:
391:
388:
385:
382:
379:
376:
373:
370:
367:
364:
359:
355:
351:
348:
345:
342:
339:
336:
333:
327:
324:
319:
316:
298:satisfies the
287:
272:Alan C. Newell
268:
267:
254:
249:
244:
240:
236:
233:
230:
227:
224:
219:
215:
211:
205:
202:
197:
194:
176:satisfies the
165:
142:
122:
102:
80:
76:
53:
49:
27:, named after
13:
10:
9:
6:
4:
3:
2:
2558:
2547:
2544:
2542:
2539:
2537:
2534:
2533:
2531:
2516:
2513:
2507:
2504:
2498:
2495:
2489:
2486:
2480:
2477:
2471:
2468:
2462:
2458:
2455:
2453:
2450:
2448:
2445:
2443:
2440:
2439:
2435:
2433:
2420:
2415:
2411:
2404:
2401:
2398:
2395:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2343:
2340:
2337:
2334:
2328:
2320:
2290:
2280:
2272:
2266:
2263:
2260:
2257:
2251:
2248:
2245:
2242:
2239:
2234:
2223:
2220:
2217:
2214:
2208:
2205:
2199:
2191:
2185:
2177:
2164:
2163:
2162:
2148:
2140:
2132:
2104:
2101:
2090:
2085:
2082:
2079:
2071:
2063:
2059:
2055:
2052:
2040:
2024:
2019:
2015:
2011:
2008:
2005:
2000:
1996:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1970:
1965:
1961:
1957:
1954:
1951:
1948:
1945:
1942:
1937:
1933:
1929:
1926:
1919:
1905:
1902:
1899:
1893:
1890:
1887:
1882:
1878:
1874:
1871:
1868:
1860:
1856:
1851:
1848:
1845:
1839:
1836:
1833:
1830:
1827:
1822:
1818:
1809:
1805:
1797:
1796:
1795:
1776:
1773:
1770:
1767:
1759:
1756:
1734:
1731:
1728:
1724:
1720:
1717:
1714:
1708:
1705:
1702:
1696:
1689:
1688:
1687:
1685:
1677:
1675:
1662:
1656:
1653:
1650:
1647:
1644:
1639:
1635:
1631:
1624:
1617:
1614:
1611:
1608:
1602:
1597:
1593:
1584:
1568:
1565:
1559:
1556:
1553:
1547:
1544:
1541:
1538:
1534:
1530:
1527:
1504:
1501:
1493:
1484:
1479:
1475:
1468:
1465:
1462:
1456:
1453:
1450:
1447:
1443:
1438:
1435:
1432:
1424:
1416:
1412:
1404:
1400:
1396:
1393:
1388:
1382:
1379:
1368:
1361:
1360:
1359:
1353:
1348:
1346:
1332:
1309:
1303:
1299:
1294:
1291:
1286:
1283:
1279:
1275:
1269:
1263:
1256:
1255:
1254:
1237:
1234:
1231:
1228:
1208:
1205:
1202:
1190:
1188:
1175:
1171:
1167:
1164:
1159:
1155:
1131:
1128:
1120:
1111:
1101:
1093:
1089:
1081:
1077:
1073:
1070:
1065:
1051:
1044:
1043:
1042:
1021:
1013:
1009:
1002:
996:
993:
990:
978:
973:
953:
948:
944:
940:
937:
926:
918:
915:
912:
906:
897:
891:
883:
866:
863:
860:
855:
844:
841:
839:
831:
814:
807:
804:
796:
792:
788:
785:
779:
768:
760:
757:
754:
748:
739:
733:
730:
722:
705:
702:
699:
694:
683:
681:
673:
665:
648:
647:
646:
629:
612:
584:
576:
573:
548:
544:
540:
537:
531:
528:
517:
510:Substituting
508:
506:
502:
486:
477:
463:
443:
435:
431:
410:
400:
392:
386:
383:
380:
377:
371:
368:
365:
362:
357:
346:
343:
340:
337:
331:
325:
317:
304:
303:
302:
301:
285:
277:
273:
252:
242:
234:
231:
228:
225:
222:
217:
209:
203:
195:
182:
181:
180:
179:
163:
154:
140:
120:
100:
78:
74:
51:
47:
38:
34:
30:
26:
19:
2515:
2506:
2497:
2488:
2479:
2470:
2307:
2044:
1793:
1681:
1519:
1357:
1324:
1194:
1146:
982:
509:
478:
427:
299:
269:
177:
155:
24:
22:
37:bifurcation
2546:Lev Landau
2530:Categories
2463:References
33:Lev Landau
2541:Mechanics
2405:α
2390:−
2387:ω
2372:λ
2344:α
2267:β
2252:−
2246:λ
2231:∇
2224:α
2203:∇
2200:⋅
2183:∂
2175:∂
2129:ξ
2098:ξ
2083:ω
2080:−
2072:⋅
2012:β
2009:−
1993:α
1990:−
1978:α
1958:−
1949:α
1943:−
1930:−
1903:ν
1875:α
1872:−
1869:α
1857:λ
1837:−
1834:α
1806:λ
1757:λ
1732:ν
1718:λ
1651:β
1648:α
1636:β
1618:β
1615:α
1560:β
1557:−
1554:α
1531:ω
1469:β
1466:−
1463:α
1454:β
1448:ω
1436:ω
1433:−
1425:⋅
1397:−
1295:
1241:∞
1232:≤
1102:⋅
1074:−
1022:⋅
941:β
938:−
932:Θ
923:∇
910:∇
907:⋅
904:Θ
901:∇
876:Θ
873:∇
864:−
852:∇
845:α
829:∂
824:Θ
821:∂
789:−
774:Θ
765:∇
752:∇
749:⋅
746:Θ
743:∇
734:α
731:−
715:Θ
712:∇
703:−
691:∇
671:∂
663:∂
610:Θ
552:Θ
464:β
444:α
387:β
372:−
354:∇
347:α
323:∂
315:∂
276:Lee Segel
232:−
214:∇
201:∂
193:∂
2520:289-313.
2436:See also
566:, where
2308:where
2120:where
1325:where
503:. The
2364:and
1603:>
1502:<
1292:tanh
1238:<
1165:>
1129:<
456:and
432:and
31:and
23:The
2532::
1505:1.
1176:3.
1132:1.
2421:.
2416:2
2412:k
2408:)
2402:i
2399:+
2396:1
2393:(
2384:i
2381:+
2378:1
2375:=
2351:k
2347:)
2341:i
2338:+
2335:1
2332:(
2329:+
2325:u
2321:=
2317:v
2291:2
2286:|
2281:B
2277:|
2273:B
2270:)
2264:i
2261:+
2258:1
2255:(
2249:B
2243:+
2240:B
2235:2
2227:)
2221:i
2218:+
2215:1
2212:(
2209:=
2206:B
2196:v
2192:+
2186:t
2178:B
2149:t
2145:u
2141:+
2137:x
2133:=
2108:)
2105:t
2102:,
2094:(
2091:B
2086:t
2076:x
2068:k
2064:i
2060:e
2056:=
2053:A
2025:.
2020:2
2016:L
2006:=
2001:2
1997:M
1987:M
1984:3
1981:+
1975:2
1971:,
1966:2
1962:L
1955:=
1952:M
1946:3
1938:2
1934:M
1927:2
1906:,
1900:=
1897:)
1894:M
1891:2
1888:+
1883:2
1879:M
1866:(
1861:2
1852:,
1849:1
1846:=
1843:)
1840:1
1831:2
1828:+
1823:2
1819:M
1815:(
1810:2
1777:M
1774:i
1771:+
1768:1
1764:)
1760:x
1753:h
1750:c
1747:e
1744:s
1740:(
1735:t
1729:i
1725:e
1721:L
1715:=
1712:)
1709:t
1706:,
1703:x
1700:(
1697:A
1663:.
1660:)
1657:3
1654:+
1645:+
1640:2
1632:2
1629:(
1625:/
1621:)
1612:+
1609:1
1606:(
1598:2
1594:k
1569:.
1566:k
1563:)
1551:(
1548:2
1545:=
1542:k
1539:d
1535:/
1528:d
1498:|
1494:k
1490:|
1485:,
1480:2
1476:k
1472:)
1460:(
1457:+
1451:=
1444:,
1439:t
1429:x
1421:k
1417:i
1413:e
1405:2
1401:k
1394:1
1389:=
1386:)
1383:t
1380:,
1376:x
1372:(
1369:A
1333:a
1310:,
1304:2
1300:x
1287:a
1284:i
1280:e
1276:=
1273:)
1270:x
1267:(
1264:A
1235:x
1229:0
1209:0
1206:=
1203:A
1172:/
1168:1
1160:2
1156:k
1125:|
1121:k
1117:|
1112:,
1106:x
1098:k
1094:i
1090:e
1082:2
1078:k
1071:1
1066:=
1063:)
1059:x
1055:(
1052:A
1026:x
1018:k
1014:i
1010:e
1006:)
1003:k
1000:(
997:f
994:=
991:A
954:.
949:3
945:R
935:)
927:2
919:R
916:+
913:R
898:2
895:(
892:+
889:]
884:2
880:)
870:(
867:R
861:R
856:2
848:[
842:=
832:t
815:R
808:,
805:R
802:)
797:2
793:R
786:1
783:(
780:+
777:)
769:2
761:R
758:+
755:R
740:2
737:(
728:]
723:2
719:)
709:(
706:R
700:R
695:2
687:[
684:=
674:t
666:R
633:)
630:A
627:(
623:g
620:r
617:a
613:=
589:|
585:A
581:|
577:=
574:R
549:i
545:e
541:R
538:=
535:)
532:t
529:,
525:x
521:(
518:A
487:A
411:2
406:|
401:A
397:|
393:A
390:)
384:i
381:+
378:1
375:(
369:A
366:+
363:A
358:2
350:)
344:i
341:+
338:1
335:(
332:=
326:t
318:A
286:A
253:2
248:|
243:A
239:|
235:A
229:A
226:+
223:A
218:2
210:=
204:t
196:A
164:A
141:A
121:A
101:A
79:c
75:k
52:c
48:k
20:.
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