862:
464:
306:
611:
79:
649:
339:
933:
202:
503:
857:{\displaystyle u=\left({\frac {2b^{2}\cosh(\theta )+2ib{\sqrt {2-b^{2}}}\sinh(\theta )}{2\cosh(\theta )-{\sqrt {2}}{\sqrt {2-b^{2}}}\cos(abx)}}-1\right)a\,e^{ia^{2}t}}
42:
in which energy concentrates in a localized and oscillatory fashion. This contradicts with the expectations derived from the corresponding linear system for
194:
1103:
71:
459:{\displaystyle u=4\arctan \left({\frac {{\sqrt {1-\omega ^{2}}}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2}}}\;x)}}\right),}
106:
92:
1154:
1134:
494:
137:
1046:
N. N. Akhmediev; V. M. Eleonskiǐ; N. E. Kulagin (1987). "First-order exact solutions of the nonlinear Schrödinger equation".
873:
144:
330:
17:
166:
ones. Standing breathers correspond to localized solutions whose amplitude vary in time (they are sometimes called
67:) in time. But also the opposite situation: oscillations in space and localized in time, is denoted as a breather.
25:
63:
The term breather originates from the characteristic that most breathers are localized in space and oscillate (
57:
301:{\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}-{\frac {\partial ^{2}u}{\partial x^{2}}}+\sin u=0,}
1008:
M. J. Ablowitz; D. J. Kaup; A. C. Newell; H. Segur (1973). "Method for solving the sine-Gordon equation".
148:
1167:
Kibler, B.; Fatome, J.; Finot, C.; Millot, G.; Dias, F.; Genty, G.; Akhmediev, N.; Dudley, J.M. (2010).
606:{\displaystyle i\,{\frac {\partial u}{\partial t}}+{\frac {\partial ^{2}u}{\partial x^{2}}}+|u|^{2}u=0,}
170:). A necessary condition for the existence of breathers in discrete lattices is that the breather main
1180:
1055:
1017:
190:
133:
125:
1142:
1071:
74:
This breather pseudospherical surface corresponds to a solution of a non-linear wave-equation.
1099:
966:
478:
121:
1188:
1063:
1025:
978:
1120:
312:
129:
1184:
1059:
1021:
1214:
637:
621:
163:
1208:
159:
43:
1075:
1029:
141:
111:
96:
70:
951:
21:
171:
46:
36:
105:
91:
78:
49:, which tends towards an even distribution of initially localized energy.
178:
167:
1150:
1067:
983:
155:
1193:
1168:
489:
Example of a breather solution for the nonlinear Schrödinger equation
175:
64:
77:
69:
39:
101:
is a swinging in time coupled kink-antikink 2-soliton solution.
965:. Note that a limiting case of the breather solution is the
185:
Example of a breather solution for the sine-Gordon equation
643:
One of the breather solutions (Kuznetsov-Ma breather) is
16:
For the component in an internal combustion engine, see
928:{\displaystyle \theta =a^{2}\,b\,{\sqrt {2-b^{2}}}\;t,}
876:
652:
506:
342:
205:
147:
that possess breather solutions. Discrete nonlinear
174:and all its multipliers are located outside of the
927:
856:
605:
458:
300:
1169:"The Peregrine soliton in nonlinear fibre optics"
497:is the dispersive partial differential equation:
1157:the nonlinearity parameter is of opposite sign.
158:structures. There are two types of breathers:
950:= 0. These breathers exist for values of the
8:
1118:Miroshnichenko A, Vasiliev A, Dmitriev S.
918:
439:
409:
385:
151:in many cases support breather solutions.
1192:
1082:Teoreticheskaya i Matematicheskaya Fizika
910:
898:
897:
893:
887:
875:
843:
835:
830:
784:
772:
765:
719:
707:
674:
664:
651:
585:
580:
571:
559:
541:
534:
511:
510:
505:
431:
419:
377:
365:
362:
341:
268:
250:
243:
231:
213:
206:
204:
1003:
1001:
999:
938:which gives breathers periodic in space
195:dispersive partial differential equation
104:
90:
1145:as the dispersive term proportional to
1041:
1039:
995:
20:. For the item of scuba apparatus, see
1094:N. N. Akhmediev; A. Ankiewicz (1997).
56:is a breather solution on a nonlinear
1096:Solitons, non-linear pulses and beams
946:when moving away from the focus time
329:An exact solution found by using the
318:a function of the spatial coordinate
7:
1048:Theoretical and Mathematical Physics
942:and approaching the uniform value
552:
538:
522:
514:
261:
247:
224:
210:
14:
132:equations. The exactly solvable
82:Pseudospherical breather surface
1121:Solitons and Soliton Collisions
1155:nonlinear Schrödinger equation
1153:solutions. In the de-focusing
1135:nonlinear Schrödinger equation
810:
798:
759:
753:
739:
733:
692:
686:
581:
572:
495:nonlinear Schrödinger equation
443:
416:
401:
392:
145:partial differential equations
138:nonlinear Schrödinger equation
1:
1137:has a nonlinearity parameter
1084:72(2): 183–196, August, 1987.
331:inverse scattering transform
18:Crankcase ventilation system
1030:10.1103/PhysRevLett.30.1262
1231:
120:A breather is a localized
15:
26:Breather (disambiguation)
1010:Physical Review Letters
624:field as a function of
110:Large amplitude moving
929:
858:
607:
481:when moving away from
473:, is periodic in time
460:
302:
128:equations or discrete
117:
102:
83:
75:
24:. For other uses, see
930:
859:
608:
461:
303:
108:
94:
81:
73:
990:References and notes
874:
650:
504:
479:decays exponentially
340:
203:
191:sine-Gordon equation
149:Hamiltonian lattices
140:are examples of one-
134:sine-Gordon equation
1185:2010NatPh...6..790K
1060:1987TMP....72..809A
1022:1973PhRvL..30.1262A
124:solution of either
1143:sign (mathematics)
1068:10.1007/BF01017105
925:
854:
603:
456:
298:
118:
103:
84:
76:
1194:10.1038/nphys1740
1147:∂u/∂x
1105:978-0-412-75450-0
1016:(25): 1262–1264.
967:Peregrine soliton
916:
814:
790:
770:
725:
566:
529:
447:
437:
383:
275:
238:
193:is the nonlinear
136:and the focusing
99:standing breather
54:discrete breather
1222:
1199:
1198:
1196:
1164:
1158:
1131:
1125:
1116:
1110:
1109:
1091:
1085:
1080:Translated from
1079:
1043:
1034:
1033:
1005:
979:Breather surface
964:
963:
934:
932:
931:
926:
917:
915:
914:
899:
892:
891:
863:
861:
860:
855:
853:
852:
848:
847:
826:
822:
815:
813:
791:
789:
788:
773:
771:
766:
742:
726:
724:
723:
708:
679:
678:
665:
612:
610:
609:
604:
590:
589:
584:
575:
567:
565:
564:
563:
550:
546:
545:
535:
530:
528:
520:
512:
465:
463:
462:
457:
452:
448:
446:
438:
436:
435:
420:
404:
384:
382:
381:
366:
363:
307:
305:
304:
299:
276:
274:
273:
272:
259:
255:
254:
244:
239:
237:
236:
235:
222:
218:
217:
207:
181:of the lattice.
126:continuous media
1230:
1229:
1225:
1224:
1223:
1221:
1220:
1219:
1205:
1204:
1203:
1202:
1166:
1165:
1161:
1132:
1128:
1117:
1113:
1106:
1093:
1092:
1088:
1045:
1044:
1037:
1007:
1006:
997:
992:
975:
961:
959:
906:
883:
872:
871:
839:
831:
780:
743:
715:
670:
666:
663:
659:
648:
647:
579:
555:
551:
537:
536:
521:
513:
502:
501:
491:
427:
405:
373:
364:
358:
338:
337:
264:
260:
246:
245:
227:
223:
209:
208:
201:
200:
187:
89:
29:
12:
11:
5:
1228:
1226:
1218:
1217:
1207:
1206:
1201:
1200:
1173:Nature Physics
1159:
1126:
1111:
1104:
1086:
1054:(2): 809–818.
1035:
994:
993:
991:
988:
987:
986:
981:
974:
971:
936:
935:
924:
921:
913:
909:
905:
902:
896:
890:
886:
882:
879:
865:
864:
851:
846:
842:
838:
834:
829:
825:
821:
818:
812:
809:
806:
803:
800:
797:
794:
787:
783:
779:
776:
769:
764:
761:
758:
755:
752:
749:
746:
741:
738:
735:
732:
729:
722:
718:
714:
711:
706:
703:
700:
697:
694:
691:
688:
685:
682:
677:
673:
669:
662:
658:
655:
638:imaginary unit
614:
613:
602:
599:
596:
593:
588:
583:
578:
574:
570:
562:
558:
554:
549:
544:
540:
533:
527:
524:
519:
516:
509:
490:
487:
467:
466:
455:
451:
445:
442:
434:
430:
426:
423:
418:
415:
412:
408:
403:
400:
397:
394:
391:
388:
380:
376:
372:
369:
361:
357:
354:
351:
348:
345:
309:
308:
297:
294:
291:
288:
285:
282:
279:
271:
267:
263:
258:
253:
249:
242:
234:
230:
226:
221:
216:
212:
186:
183:
154:Breathers are
88:
85:
31:In physics, a
13:
10:
9:
6:
4:
3:
2:
1227:
1216:
1213:
1212:
1210:
1195:
1190:
1186:
1182:
1178:
1174:
1170:
1163:
1160:
1156:
1152:
1148:
1144:
1140:
1136:
1133:The focusing
1130:
1127:
1123:
1122:
1115:
1112:
1107:
1101:
1097:
1090:
1087:
1083:
1077:
1073:
1069:
1065:
1061:
1057:
1053:
1049:
1042:
1040:
1036:
1031:
1027:
1023:
1019:
1015:
1011:
1004:
1002:
1000:
996:
989:
985:
982:
980:
977:
976:
972:
970:
968:
957:
953:
949:
945:
941:
922:
919:
911:
907:
903:
900:
894:
888:
884:
880:
877:
870:
869:
868:
849:
844:
840:
836:
832:
827:
823:
819:
816:
807:
804:
801:
795:
792:
785:
781:
777:
774:
767:
762:
756:
750:
747:
744:
736:
730:
727:
720:
716:
712:
709:
704:
701:
698:
695:
689:
683:
680:
675:
671:
667:
660:
656:
653:
646:
645:
644:
641:
639:
635:
631:
627:
623:
619:
600:
597:
594:
591:
586:
576:
568:
560:
556:
547:
542:
531:
525:
517:
507:
500:
499:
498:
496:
493:The focusing
488:
486:
484:
480:
476:
472:
453:
449:
440:
432:
428:
424:
421:
413:
410:
406:
398:
395:
389:
386:
378:
374:
370:
367:
359:
355:
352:
349:
346:
343:
336:
335:
334:
332:
327:
325:
321:
317:
314:
295:
292:
289:
286:
283:
280:
277:
269:
265:
256:
251:
240:
232:
228:
219:
214:
199:
198:
197:
196:
192:
184:
182:
180:
177:
173:
169:
165:
161:
157:
152:
150:
146:
143:
139:
135:
131:
127:
123:
115:
113:
107:
100:
98:
93:
86:
80:
72:
68:
66:
61:
59:
55:
50:
48:
45:
44:infinitesimal
41:
38:
34:
27:
23:
19:
1176:
1172:
1162:
1146:
1141:of the same
1138:
1129:
1119:
1114:
1098:. Springer.
1095:
1089:
1081:
1051:
1047:
1013:
1009:
955:
947:
943:
939:
937:
866:
642:
636:denotes the
633:
629:
625:
617:
615:
492:
482:
474:
470:
468:
328:
323:
319:
315:
310:
188:
153:
119:
109:
95:
62:
53:
51:
32:
30:
1179:(10): 790.
469:which, for
142:dimensional
112:sine-Gordon
97:Sine-Gordon
1149:, and has
958:less than
954:parameter
952:modulation
632:. Further
47:amplitudes
22:rebreather
904:−
878:θ
817:−
796:
778:−
763:−
757:θ
751:
737:θ
731:
713:−
690:θ
684:
553:∂
539:∂
523:∂
515:∂
429:ω
425:−
414:
407:ω
396:ω
390:
375:ω
371:−
356:
322:and time
311:with the
284:
262:∂
248:∂
241:−
225:∂
211:∂
172:frequency
168:oscillons
164:traveling
156:solitonic
37:nonlinear
1209:Category
1076:18571794
973:See also
471:ω < 1
179:spectrum
160:standing
122:periodic
114:breather
87:Overview
33:breather
1181:Bibcode
1151:soliton
1056:Bibcode
1018:Bibcode
984:Soliton
960:√
622:complex
130:lattice
65:breathe
58:lattice
1102:
1074:
353:arctan
176:phonon
1215:Waves
1072:S2CID
867:with
616:with
483:x = 0
313:field
35:is a
1100:ISBN
748:cosh
728:sinh
681:cosh
628:and
477:and
411:cosh
333:is:
189:The
40:wave
1189:doi
1064:doi
1026:doi
793:cos
387:cos
281:sin
162:or
1211::
1187:.
1175:.
1171:.
1070:.
1062:.
1052:72
1050:.
1038:^
1024:.
1014:30
1012:.
998:^
969:.
640:.
620:a
485:.
326:.
60:.
52:A
1197:.
1191::
1183::
1177:6
1139:κ
1124:.
1108:.
1078:.
1066::
1058::
1032:.
1028::
1020::
962:2
956:b
948:t
944:a
940:x
923:,
920:t
912:2
908:b
901:2
895:b
889:2
885:a
881:=
850:t
845:2
841:a
837:i
833:e
828:a
824:)
820:1
811:)
808:x
805:b
802:a
799:(
786:2
782:b
775:2
768:2
760:)
754:(
745:2
740:)
734:(
721:2
717:b
710:2
705:b
702:i
699:2
696:+
693:)
687:(
676:2
672:b
668:2
661:(
657:=
654:u
634:i
630:t
626:x
618:u
601:,
598:0
595:=
592:u
587:2
582:|
577:u
573:|
569:+
561:2
557:x
548:u
543:2
532:+
526:t
518:u
508:i
475:t
454:,
450:)
444:)
441:x
433:2
422:1
417:(
402:)
399:t
393:(
379:2
368:1
360:(
350:4
347:=
344:u
324:t
320:x
316:u
296:,
293:0
290:=
287:u
278:+
270:2
266:x
257:u
252:2
233:2
229:t
220:u
215:2
116:.
28:.
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