520:
1037:
95:
20:
1090:
1816:
components, it has been shown that even with an approximate initial condition (in the case of this work, an initial sinusoidal beating), a profile very close to the ideal
Peregrine soliton can be generated. However, the non-ideal input condition lead to substructures that appear after the point of
1101:. In fact, the evolution of light in fiber optics and the evolution of surface waves in deep water are both modelled by the nonlinear Schrödinger equation (note however that spatial and temporal variables have to be switched). Such an analogy has been exploited in the past in order to generate
73:
localization. Therefore, starting from a weak oscillation on a continuous background, the
Peregrine soliton develops undergoing a progressive increase of its amplitude and a narrowing of its temporal duration. At the point of maximum compression, the amplitude is three times the level of the
1237:
811:
223:
978:
74:
continuous background (and if one considers the intensity as it is relevant in optics, there is a factor 9 between the peak intensity and the surrounding background). After this point of maximal compression, the wave's amplitude decreases and its width increases.
1113:
552:
1606:
109:
1097:
In 2010, more than 25 years after the initial work of
Peregrine, researchers took advantage of the analogy that can be drawn between hydrodynamics and optics in order to generate Peregrine solitons in
849:
1696:
454:
1807:
81:. Therefore, the Peregrine soliton is an attractive hypothesis to explain the formation of those waves which have a high amplitude and may appear from nowhere and disappear without a trace.
302:
1299:
1266:
1997:
1319:
1026:
833:
509:
1639:
483:
1750:
1723:
1000:
545:
267:
2212:
Erkintalo, M.; Kibler, B.; Hammani, K.; Finot, C.; Akhmediev, N.; Dudley, J.M.; Genty, G. (2011). "Higher-Order
Modulation Instability in Nonlinear Fiber Optics".
247:
1359:
1339:
2016:
Kibler, B.; Fatome, J.; Finot, C.; Millot, G.; Dias, F.; Genty, G.; Akhmediev, N.; Dudley, J.M. (2010). "The
Peregrine soliton in nonlinear fibre optics".
102:
The
Peregrine soliton is a solution of the one-dimensional nonlinear Schrödinger equation that can be written in normalized units as follows :
1232:{\displaystyle i{\frac {\partial \psi }{\partial z}}-{\frac {\beta _{2}}{2}}{\frac {\partial ^{2}\psi }{\partial t^{2}}}+\gamma |\psi |^{2}\psi =0}
1836:. In 2013, complementary experiments using a scale model of a chemical tanker ship have discussed the potential devastating effects on the ship.
806:{\displaystyle {\tilde {\psi }}(\eta ,\tau )={\frac {1}{\sqrt {2\pi }}}\int {\psi (\xi ,\tau )e^{i\eta \xi }d\xi }={\sqrt {2\pi }}e^{i\tau }\left}
1108:
More precisely, the nonlinear Schrödinger equation can be written in the context of optical fibers under the following dimensional form :
1002:, the modulus of the spectrum exhibits a typical triangular shape when plotted on a logarithmic scale. The broadest spectrum is obtained for
1370:
1069:
The
Peregrine soliton can also be seen as the limiting case of the time-periodic Kuznetsov-Ma breather when the period tends to infinity.
1081:. This is however very different from where the Peregrine soliton has been for the first time experimentally generated and characterized.
218:{\displaystyle i{\frac {\partial \psi }{\partial \tau }}+{\frac {1}{2}}{\frac {\partial ^{2}\psi }{\partial \xi ^{2}}}+|\psi |^{2}\psi =0}
77:
These features of the
Peregrine soliton are fully consistent with the quantitative criteria usually used in order to qualify a wave as a
1817:
maximum compression. Those substructures have also a profile close to a
Peregrine soliton, which can be analytically explained using a
2470:
Bailung, H.; Sharma, S. K.; Nakamura, Y. (2011). "Observation of
Peregrine solitons in a multicomponent plasma with negative ions".
1857:
46:
973:{\displaystyle |{\tilde {\psi }}(\eta ,\tau )|={\sqrt {2\pi }}\exp \left(-{\frac {|\eta |}{2}}{\sqrt {1+4\tau ^{2}}}\right).}
519:
1929:
Shrira, V.I.; Geogjaev, V.V. (2009). "What makes the Peregrine soliton so special as a prototype of freak waves ?".
1036:
316:(note that similar results could be obtained for a normally dispersive medium combined with a defocusing nonlinearity).
1848:
have also highlighted the emergence of Peregrine solitons in other fields ruled by the nonlinear Schrödinger equation.
1832:
These results in optics have been confirmed in 2011 in hydrodynamics with experiments carried out in a 15-m long water
1644:
325:
1755:
2534:
2386:
2266:
2079:
2524:
70:
2265:
Hammani K.; Wetzel B.; Kibler B.; Fatome J.; Finot C.; Millot G.; Akhmediev N. & Dudley J. M. (2011).
2033:
305:
1991:
1813:
54:
2163:
527:
It is also possible to mathematically express the Peregrine soliton according to the spatial frequency
272:
2539:
2479:
2424:
2334:
2281:
2221:
2178:
2094:
2025:
1963:
1271:
836:
2325:
Chabchoub, A.; Hoffmann, N.P.; Akhmediev, N. (2011). "Rogue wave observation in a water wave tank".
2038:
69:
that can maintain its profile unchanged during propagation, the Peregrine soliton presents a double
1873:
1818:
309:
2368:
2194:
2051:
98:
Spatial and temporal profiles of a Peregrine soliton obtained at the point of maximum compression
1948:
1244:
2519:
2495:
2452:
2408:
2360:
2307:
2247:
2120:
1089:
1057:
The Peregrine soliton can also be seen as the limiting case of the space-periodic Akhmediev
1028:, which corresponds to the maximum of compression of the spatio-temporal nonlinear structure.
42:
1304:
1005:
818:
488:
2487:
2442:
2432:
2350:
2342:
2297:
2289:
2237:
2229:
2186:
2110:
2102:
2078:
Hammani, K.; Kibler, B.; Finot, C.; Morin, P.; Fatome, J.; Dudley, J.M.; Millot, G. (2011).
2043:
1979:
1971:
1909:
1845:
1614:
1102:
462:
50:
1728:
1701:
985:
530:
252:
94:
232:
1077:
Mathematical predictions by H. Peregrine had initially been established in the domain of
2483:
2428:
2338:
2285:
2225:
2182:
2098:
2029:
1967:
2529:
2447:
2412:
2267:"Spectral dynamics of modulation instability described using Akhmediev breather theory"
2159:
1344:
1324:
843:
2513:
2198:
1098:
1078:
313:
2372:
2055:
19:
2491:
2346:
2233:
2190:
1975:
2437:
1364:
In this context, the Peregrine soliton has the following dimensional expression:
2413:"Rogue Waves: From Nonlinear Schrödinger Breather Solutions to Sea-Keeping Test"
2164:"Akhmediev breather evolution in optical fiber for realistic initial conditions"
2080:"Peregrine soliton generation and breakup in standard telecommunications fiber"
1914:
1897:
1867:
1833:
1824:
The typical triangular spectral shape has also been experimentally confirmed.
78:
2138:
2499:
2456:
2364:
2311:
2251:
2124:
2115:
1947:
Akhmediev, N., Ankiewicz, A., Soto-Crespo, J. M. and Dudley J. M. (2011).
2293:
2106:
1862:
1058:
37:
1983:
1361:
are the propagation distance and the temporal coordinate respectively.
66:
30:
2355:
2302:
2242:
2047:
1601:{\displaystyle \psi (z,t)={\sqrt {P_{0}}}\lefte^{\dfrac {iz}{L_{NL}}}}
1898:"Water waves, nonlinear Schrödinger equations and their solutions"
1268:
being the second order dispersion (supposed to be anomalous, i.e.
1088:
1035:
518:
93:
18:
1093:
Record of the temporal profile of a Peregrine soliton in optics
1949:"Universal triangular spectra in parametrically-driven systems"
23:
3D view of the spatio-temporal evolution of a Peregrine soliton
846:(with the constant continuous background here omitted) :
1049:
The Peregrine soliton is a first-order rational soliton.
459:
so that the temporal and spatial maxima are obtained for
16:
Analytic solution of the nonlinear Schrödinger equation
1758:
1731:
1704:
1665:
1647:
1617:
1571:
1527:
1487:
1444:
1373:
1347:
1327:
1307:
1274:
1247:
1116:
1008:
988:
852:
821:
555:
533:
491:
465:
328:
275:
255:
235:
112:
523:Evolution of the spectrum of the Peregrine soliton
1801:
1744:
1717:
1690:
1633:
1600:
1353:
1333:
1313:
1293:
1260:
1231:
1032:Different interpretations of the Peregrine soliton
1020:
994:
972:
827:
805:
539:
503:
477:
448:
296:
261:
241:
217:
53:, researcher at the mathematics department of the
1691:{\displaystyle L_{NL}={\dfrac {1}{\gamma P_{0}}}}
449:{\displaystyle \psi (\xi ,\tau )=\lefte^{i\tau }}
1802:{\displaystyle T_{0}={\sqrt {\beta _{2}L_{NL}}}}
2162:; Genty, G.; Wetzel, B.; Dudley, J. M. (2011).
1942:
1940:
1725:being the power of the continuous background.
1040:Peregrine soliton and other nonlinear solutions
8:
2073:
2071:
2069:
2067:
2065:
1996:: CS1 maint: multiple names: authors list (
2011:
2009:
2007:
1891:
1889:
2446:
2436:
2354:
2301:
2241:
2114:
2037:
1913:
1788:
1778:
1772:
1763:
1757:
1736:
1730:
1709:
1703:
1678:
1664:
1652:
1646:
1622:
1616:
1585:
1570:
1552:
1536:
1526:
1509:
1496:
1486:
1453:
1443:
1420:
1401:
1395:
1372:
1346:
1326:
1306:
1279:
1273:
1252:
1246:
1214:
1209:
1200:
1185:
1167:
1160:
1149:
1143:
1120:
1115:
1007:
987:
954:
939:
928:
920:
917:
893:
885:
859:
858:
853:
851:
820:
770:
755:
744:
736:
733:
710:
677:
663:
649:
627:
604:
586:
557:
556:
554:
532:
490:
464:
437:
419:
403:
361:
327:
274:
254:
234:
200:
195:
186:
174:
156:
149:
139:
116:
111:
2139:"Peregrine's 'Soliton' observed at last"
319:The Peregrine analytical expression is:
49:. This solution was proposed in 1983 by
1885:
1321:being the nonlinear Kerr coefficient.
982:One can notice that for any given time
1989:
1844:Other experiments carried out in the
1840:Generation in other fields of physics
312:is anomalous and the nonlinearity is
308:of a surface wave in deep water. The
7:
1061:when the period tends to infinity.
1178:
1164:
1131:
1123:
167:
153:
127:
119:
65:Contrary to the usual fundamental
14:
1641:is a nonlinear length defined as
297:{\displaystyle \psi (\xi ,\tau )}
1294:{\displaystyle \beta _{2}<0}
2492:10.1103/physrevlett.107.255005
2347:10.1103/PhysRevLett.106.204502
2234:10.1103/PhysRevLett.107.253901
2191:10.1016/j.physleta.2011.04.002
1976:10.1016/j.physleta.2010.11.044
1858:Nonlinear Schrödinger equation
1812:By using exclusively standard
1389:
1377:
1210:
1201:
929:
921:
886:
882:
870:
864:
854:
795:
789:
745:
737:
620:
608:
580:
568:
562:
385:
367:
344:
332:
291:
279:
196:
187:
47:nonlinear Schrödinger equation
1:
90:In the spatio-temporal domain
2438:10.1371/journal.pone.0054629
1828:Generation in hydrodynamics
249:the spatial coordinate and
2556:
2407:Onorato, M.; Proment, D.;
1752:is a duration defined as
1261:{\displaystyle \beta _{2}}
1073:Experimental demonstration
1915:10.1017/S0334270000003891
1896:Peregrine, D. H. (1983).
1065:As a Kuznetsov-Ma soliton
269:the temporal coordinate.
1053:As an Akhmediev breather
2214:Physical Review Letters
1314:{\displaystyle \gamma }
1021:{\displaystyle \tau =0}
828:{\displaystyle \delta }
504:{\displaystyle \tau =0}
85:Mathematical expression
2387:"Rogue waves captured"
1803:
1746:
1719:
1692:
1635:
1634:{\displaystyle L_{NL}}
1602:
1355:
1335:
1315:
1295:
1262:
1233:
1094:
1041:
1022:
996:
974:
842:This corresponds to a
829:
807:
541:
524:
515:In the spectral domain
505:
479:
478:{\displaystyle \xi =0}
450:
298:
263:
243:
219:
99:
24:
2411:; Clauss, M. (2013).
2389:. www.sciencenews.org
2280:(2140–2142): 2140–2.
1902:J. Austral. Math. Soc
1814:optical communication
1804:
1747:
1745:{\displaystyle T_{0}}
1720:
1718:{\displaystyle P_{0}}
1693:
1636:
1603:
1356:
1336:
1316:
1296:
1263:
1234:
1092:
1045:As a rational soliton
1039:
1023:
997:
995:{\displaystyle \tau }
975:
830:
808:
542:
540:{\displaystyle \eta }
522:
506:
480:
451:
299:
264:
262:{\displaystyle \tau }
244:
220:
97:
55:University of Bristol
22:
2294:10.1364/OL.36.002140
2107:10.1364/OL.36.000112
1880:Notes and references
1756:
1729:
1702:
1645:
1615:
1371:
1345:
1325:
1305:
1272:
1245:
1114:
1085:Generation in optics
1006:
986:
850:
837:Dirac delta function
819:
553:
531:
489:
463:
326:
273:
253:
242:{\displaystyle \xi }
233:
110:
2484:2011PhRvL.107y5005B
2429:2013PLoSO...854629O
2339:2011PhRvL.106t4502C
2286:2011OptL...36.2140H
2226:2011PhRvL.107y3901E
2183:2011PhLA..375.2029E
2099:2011OptL...36..112H
2030:2010NatPh...6..790K
1968:2011PhLA..375..775A
1874:Optical rogue waves
1105:in optical fibers.
1846:physics of plasmas
1799:
1742:
1715:
1688:
1686:
1631:
1598:
1595:
1546:
1503:
1463:
1351:
1331:
1311:
1291:
1258:
1229:
1095:
1042:
1018:
992:
970:
825:
803:
537:
525:
501:
475:
446:
294:
259:
239:
215:
100:
25:
2177:(19): 2029–2034.
2048:10.1038/nphys1740
1797:
1685:
1594:
1559:
1545:
1502:
1462:
1407:
1354:{\displaystyle t}
1334:{\displaystyle z}
1192:
1158:
1138:
960:
937:
901:
867:
776:
753:
717:
716:
657:
599:
598:
565:
426:
181:
147:
134:
43:analytic solution
2547:
2535:Nonlinear optics
2504:
2503:
2467:
2461:
2460:
2450:
2440:
2404:
2398:
2397:
2395:
2394:
2383:
2377:
2376:
2358:
2322:
2316:
2315:
2305:
2271:
2262:
2256:
2255:
2245:
2209:
2203:
2202:
2168:
2156:
2150:
2149:
2147:
2146:
2135:
2129:
2128:
2118:
2084:
2075:
2060:
2059:
2041:
2013:
2002:
2001:
1995:
1987:
1953:
1944:
1935:
1934:
1926:
1920:
1919:
1917:
1893:
1821:transformation.
1808:
1806:
1805:
1800:
1798:
1796:
1795:
1783:
1782:
1773:
1768:
1767:
1751:
1749:
1748:
1743:
1741:
1740:
1724:
1722:
1721:
1716:
1714:
1713:
1697:
1695:
1694:
1689:
1687:
1684:
1683:
1682:
1666:
1660:
1659:
1640:
1638:
1637:
1632:
1630:
1629:
1607:
1605:
1604:
1599:
1597:
1596:
1593:
1592:
1580:
1572:
1565:
1561:
1560:
1558:
1557:
1556:
1551:
1547:
1544:
1543:
1528:
1514:
1513:
1508:
1504:
1501:
1500:
1488:
1470:
1469:
1465:
1464:
1461:
1460:
1445:
1421:
1408:
1406:
1405:
1396:
1360:
1358:
1357:
1352:
1340:
1338:
1337:
1332:
1320:
1318:
1317:
1312:
1300:
1298:
1297:
1292:
1284:
1283:
1267:
1265:
1264:
1259:
1257:
1256:
1238:
1236:
1235:
1230:
1219:
1218:
1213:
1204:
1193:
1191:
1190:
1189:
1176:
1172:
1171:
1161:
1159:
1154:
1153:
1144:
1139:
1137:
1129:
1121:
1103:optical solitons
1027:
1025:
1024:
1019:
1001:
999:
998:
993:
979:
977:
976:
971:
966:
962:
961:
959:
958:
940:
938:
933:
932:
924:
918:
902:
894:
889:
869:
868:
860:
857:
834:
832:
831:
826:
812:
810:
809:
804:
802:
798:
782:
778:
777:
775:
774:
756:
754:
749:
748:
740:
734:
718:
715:
714:
696:
695:
678:
671:
670:
658:
650:
645:
638:
637:
600:
591:
587:
567:
566:
558:
546:
544:
543:
538:
510:
508:
507:
502:
484:
482:
481:
476:
455:
453:
452:
447:
445:
444:
432:
428:
427:
425:
424:
423:
408:
407:
388:
362:
303:
301:
300:
295:
268:
266:
265:
260:
248:
246:
245:
240:
224:
222:
221:
216:
205:
204:
199:
190:
182:
180:
179:
178:
165:
161:
160:
150:
148:
140:
135:
133:
125:
117:
51:Howell Peregrine
2555:
2554:
2550:
2549:
2548:
2546:
2545:
2544:
2510:
2509:
2508:
2507:
2472:Phys. Rev. Lett
2469:
2468:
2464:
2406:
2405:
2401:
2392:
2390:
2385:
2384:
2380:
2327:Phys. Rev. Lett
2324:
2323:
2319:
2269:
2264:
2263:
2259:
2211:
2210:
2206:
2166:
2158:
2157:
2153:
2144:
2142:
2137:
2136:
2132:
2082:
2077:
2076:
2063:
2039:10.1.1.222.8599
2024:(10): 790–795.
2015:
2014:
2005:
1988:
1951:
1946:
1945:
1938:
1928:
1927:
1923:
1895:
1894:
1887:
1882:
1854:
1842:
1830:
1784:
1774:
1759:
1754:
1753:
1732:
1727:
1726:
1705:
1700:
1699:
1674:
1670:
1648:
1643:
1642:
1618:
1613:
1612:
1581:
1573:
1566:
1532:
1522:
1521:
1492:
1482:
1481:
1471:
1449:
1430:
1426:
1422:
1413:
1409:
1397:
1369:
1368:
1343:
1342:
1323:
1322:
1303:
1302:
1275:
1270:
1269:
1248:
1243:
1242:
1208:
1181:
1177:
1163:
1162:
1145:
1130:
1122:
1112:
1111:
1087:
1075:
1067:
1055:
1047:
1034:
1004:
1003:
984:
983:
950:
919:
913:
909:
848:
847:
817:
816:
766:
735:
729:
725:
706:
679:
676:
672:
659:
623:
551:
550:
529:
528:
517:
487:
486:
461:
460:
433:
415:
399:
389:
363:
354:
350:
324:
323:
271:
270:
251:
250:
231:
230:
194:
170:
166:
152:
151:
126:
118:
108:
107:
92:
87:
71:spatio-temporal
63:
61:Main properties
17:
12:
11:
5:
2553:
2551:
2543:
2542:
2537:
2532:
2527:
2525:Fluid dynamics
2522:
2512:
2511:
2506:
2505:
2478:(25): 255005.
2462:
2399:
2378:
2333:(20): 204502.
2317:
2257:
2220:(25): 253901.
2204:
2151:
2130:
2116:2027.42/149759
2093:(2): 112–114.
2087:Optics Letters
2061:
2018:Nature Physics
2003:
1962:(3): 775–779.
1936:
1921:
1884:
1883:
1881:
1878:
1877:
1876:
1871:
1865:
1860:
1853:
1850:
1841:
1838:
1829:
1826:
1794:
1791:
1787:
1781:
1777:
1771:
1766:
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