2793:
2523:
22:
2788:{\displaystyle \left({\frac {10630+674\gamma +695.2419\gamma ^{2}+191.4489\gamma ^{3}+16.86221\gamma ^{4}+4.082607\gamma ^{5}+\gamma ^{6}}{10630+674\gamma +2467\gamma ^{2}+303.2428\gamma ^{3}+164.6842\gamma ^{4}+36.6434\gamma ^{5}+3.9596\gamma ^{6}+0.8983\gamma ^{7}+{\frac {16}{\pi ^{4}}}\gamma ^{8}}}\right)^{1/4}}
2954:
give additive contributions to the decay; the decay rate, as well as the amplitude and phase of the nonlinear oscillation, can be approximated with elementary functions in a manner similar to the period above. In describing the behavior of the idealized Toda oscillator, the error of such
2986:
is achieved by introducing a new spatial coordinate which is independent of the chain index label. This is done in a relativistically invariant way, so that time and space are treated on equal grounds. This means that the Toda field theory is not a continuous limit of the Toda chain.
325:
1545:
1954:
1360:
2175:
2517:
2304:
768:
1859:
658:
2439:
1806:
1751:
949:
445:
1446:
199:
2840:
1684:
385:
2375:
2338:
2209:
2076:
40:
1864:
1614:
1581:
556:
2041:
1980:
1177:
1062:
1222:
1263:
1141:
589:
3213:
2952:
2926:
2900:
2874:
1640:
1438:
1412:
1386:
1268:
1092:
1012:
979:
886:
860:
834:
801:
688:
509:
479:
191:
161:
135:
2003:
2081:
2447:
3149:
1097:
Both analogies, with a unity mass particle and logarithm of number of photons, are useful in the analysis of behavior of the Toda oscillator.
3092:
Kouznetsov, D.; Bisson, J.-F.; Li, J.; Ueda, K. (2007). "Self-pulsing laser as Toda oscillator: Approximation through elementary functions".
2214:
693:
862:
are supposed to be positive; then this speed-proportional friction coefficient grows exponentially at large positive values of coordinate
58:
1811:
600:
83:. It represents a chain of particles with exponential potential interaction between neighbors. These concepts are named after
2976:
2380:
1756:
3208:
1143:. Indeed, in the realization of the Toda oscillator as a self-pulsing laser, these parameters may have values of order of
2902:, the pulsation decays slowly, and this decay can be described analytically. In the first approximation, the parameters
3046:
1699:
1753:
is a convenient parameter to characterize the amplitude of pulsation. Using this, we can express the median value
1179:; during several pulses, the amplitude of pulsation does not change much. In this case, we can speak about the
2975:
equations of motion, in the continuous limit in which the distance between neighbors goes to zero, become the
1540:{\displaystyle z=\pm \int _{x_{\min }}^{x_{\max }}\!\!{\frac {{\rm {d}}a}{{\sqrt {2}}{\sqrt {E-\Phi (a)}}}}}
3111:
1690:
894:
390:
320:{\displaystyle {\frac {{\rm {d^{2}}}x}{{\rm {d}}z^{2}}}+D(x){\frac {{\rm {d}}x}{{\rm {d}}z}}+\Phi '(x)=0,}
80:
3147:
Kashaev, R.-M.; Reshetikhin, N. (1997). "Affine Toda field theory as a 3-dimensional integrable system".
2005:
need not be the physical energy of the system; in these cases, this dimensionless quantity may be called
1388:, and can be treated as a constant of motion. Then, during one period of pulsation, the relation between
2955:
approximations is smaller than the differences between the ideal and its experimental realization as a
2979:(KdV) equation. Here the index labeling the particle in the chain becomes the new spatial coordinate.
2809:
96:
3168:
3103:
3094:
3055:
3018:
1949:{\displaystyle E=E(\gamma )={\frac {\gamma }{\tanh(\gamma )}}+\ln {\frac {\sinh \gamma }{\gamma }}-1}
1645:
333:
3116:
2343:
2311:
2182:
2049:
2803:
2799:
1023:
3184:
3158:
3129:
3071:
2006:
1586:
1553:
514:
2020:
1959:
1146:
1029:
2983:
1186:
92:
1230:
1108:
561:
3176:
3121:
3063:
3026:
1355:{\displaystyle ~E={\frac {1}{2}}\left({\frac {{\rm {d}}x}{{\rm {d}}z}}\right)^{2}+\Phi (x)~}
952:
108:
3009:
2931:
2905:
2879:
2853:
2170:{\displaystyle ~T(\gamma )=2\pi \left(1+{\frac {\gamma ^{2}}{24}}+O(\gamma ^{4})\right)~}
1619:
1417:
1391:
1365:
1071:
991:
958:
865:
839:
813:
780:
667:
488:
458:
170:
140:
114:
3172:
3107:
3059:
3022:
1988:
88:
3202:
3125:
3075:
3030:
2960:
985:
84:
3188:
3133:
2972:
2956:
1019:
771:
2512:{\displaystyle k_{\text{fit}}(\gamma )={\frac {2\pi }{T_{\text{fit}}(\gamma )}}=}
87:. The Toda oscillator is used as a simple model to understand the phenomenon of
1065:
595:
2963:. However, a self-pulsing laser shows qualitatively very similar behavior.
1180:
1015:
164:
3180:
804:
661:
3163:
3067:
72:
2299:{\displaystyle ~T(\gamma )=4\gamma ^{1/2}\left(1+O(1/\gamma )\right)~}
763:{\displaystyle ~{\ddot {x}}={\frac {{\rm {d}}^{2}x}{{\rm {d}}z^{2}}}~}
3044:
Oppo, G.L.; Politi, A. (1985). "Toda potential in laser equations".
2017:
The period of pulsation is an increasing function of the amplitude
91:, which is a quasi-periodic pulsation of the output intensity of a
1689:
however, other solutions may be obtained using the principle of
482:
111:
of any origin, which can be described with dependent coordinate
15:
1854:{\displaystyle \delta =\ln {\frac {\sin(\gamma )}{\gamma }}}
653:{\displaystyle ~{\dot {x}}={\frac {{\rm {d}}x}{{\rm {d}}z}}}
2434:{\displaystyle ~k(\gamma )={\frac {2\pi }{T(\gamma )}}~}
1801:{\displaystyle \delta ={\frac {x_{\max }-x_{\min }}{1}}}
803:
may have sense of coefficient of the speed-proportional
36:
2934:
2908:
2882:
2856:
2812:
2526:
2450:
2383:
2346:
2314:
2217:
2185:
2084:
2052:
2023:
1991:
1962:
1867:
1814:
1759:
1702:
1648:
1622:
1589:
1556:
1449:
1420:
1394:
1368:
1271:
1233:
1189:
1149:
1111:
1074:
1032:
994:
961:
897:
868:
842:
816:
783:
774:; and the mass of such a particle is equal to unity.
696:
670:
603:
564:
517:
491:
461:
393:
336:
202:
173:
143:
117:
3007:
Toda, M. (1975). "Studies of a non-linear lattice".
31:
may be too technical for most readers to understand
2946:
2920:
2894:
2868:
2834:
2787:
2511:
2433:
2369:
2332:
2298:
2203:
2169:
2070:
2035:
1997:
1974:
1948:
1853:
1800:
1745:
1678:
1634:
1608:
1575:
1539:
1432:
1406:
1380:
1354:
1257:
1216:
1171:
1135:
1086:
1056:
1006:
973:
943:
880:
854:
828:
795:
762:
682:
652:
583:
550:
503:
473:
439:
379:
319:
185:
155:
129:
1489:
1488:
1787:
1774:
1726:
1711:
1598:
1565:
1481:
1469:
1105:Rigorously, the oscillation is periodic only at
1022:, related to its steady-state value. Then, the
1746:{\displaystyle ~x_{\max }/x_{\min }=2\gamma ~}
1642:; this solution is written for the case when
8:
3162:
3115:
2933:
2907:
2881:
2855:
2823:
2811:
2775:
2771:
2758:
2746:
2737:
2728:
2712:
2696:
2680:
2664:
2648:
2618:
2605:
2589:
2573:
2557:
2532:
2525:
2488:
2473:
2455:
2449:
2402:
2382:
2350:
2345:
2313:
2277:
2247:
2243:
2216:
2184:
2150:
2126:
2120:
2083:
2051:
2022:
1990:
1961:
1922:
1889:
1866:
1827:
1813:
1786:
1773:
1766:
1758:
1725:
1716:
1710:
1701:
1650:
1649:
1647:
1621:
1597:
1588:
1564:
1555:
1512:
1505:
1494:
1493:
1490:
1480:
1475:
1468:
1463:
1448:
1419:
1393:
1367:
1328:
1312:
1311:
1300:
1299:
1296:
1281:
1270:
1232:
1188:
1157:
1148:
1110:
1073:
1031:
993:
960:
920:
896:
867:
841:
815:
782:
748:
738:
737:
726:
720:
719:
715:
701:
700:
695:
669:
638:
637:
626:
625:
622:
608:
607:
602:
572:
563:
539:
530:
516:
490:
485:. Indeed, it may be proportional to time
460:
416:
392:
362:
335:
276:
275:
264:
263:
260:
236:
226:
225:
212:
207:
206:
203:
201:
172:
142:
116:
59:Learn how and when to remove this message
43:, without removing the technical details.
2850:At small (but still positive) values of
2996:
955:at large positive values of coordinate
3214:Atomic, molecular, and optical physics
3150:Communications in Mathematical Physics
3087:
3085:
3002:
3000:
2806:of this approximation does not exceed
951:is a fixed function, which also shows
41:make it understandable to non-experts
7:
944:{\displaystyle ~\Phi (x)=e^{x}-x-1~}
440:{\displaystyle ~\Phi (x)=e^{x}-x-1~}
1026:of such a laser is proportional to
1956:is also an elementary function of
1616:are minimal and maximal values of
1520:
1495:
1337:
1313:
1301:
901:
739:
721:
639:
627:
447:and prime denotes the derivative.
397:
292:
277:
265:
227:
209:
193:can be approximated with equation
14:
1183:of pulsation, since the function
2835:{\displaystyle 22\times 10^{-9}}
20:
1679:{\displaystyle {\dot {x}}(0)=0}
1440:can be expressed analytically:
1265:, the energy of the oscillator
380:{\displaystyle ~D(x)=ue^{x}+v~}
2500:
2494:
2467:
2461:
2422:
2416:
2396:
2390:
2370:{\displaystyle ~{T(\gamma )}~}
2360:
2354:
2333:{\displaystyle ~\gamma >0~}
2285:
2271:
2230:
2224:
2204:{\displaystyle ~\gamma \gg 1~}
2156:
2143:
2097:
2091:
2071:{\displaystyle ~\gamma \ll 1~}
1907:
1901:
1883:
1877:
1842:
1836:
1667:
1661:
1529:
1523:
1346:
1340:
1208:
1202:
1048:
1042:
910:
904:
406:
400:
349:
343:
305:
299:
257:
251:
1:
1985:In application, the quantity
167:along independent coordinate
3031:10.1016/0370-1573(75)90018-6
1018:of number of photons in the
664:of particle with coordinate
163:, characterized in that the
1609:{\displaystyle ~x_{\max }~}
1576:{\displaystyle ~x_{\min }~}
551:{\displaystyle ~z=t/t_{0}~}
455:The independent coordinate
137:and independent coordinate
3230:
3126:10.1088/1751-8113/40/9/016
2977:Korteweg–de Vries equation
1064:and may show pulsation at
2036:{\displaystyle ~\gamma ~}
1975:{\displaystyle ~\gamma ~}
1172:{\displaystyle ~10^{-4}~}
1057:{\displaystyle ~\exp(x)~}
810:Usually, both parameters
777:The dissipative function
107:The Toda oscillator is a
3047:Zeitschrift für Physik B
1691:translational invariance
1217:{\displaystyle ~x=x(t)~}
511:with some relation like
2441:can be approximated by
1258:{\displaystyle ~u=v=0~}
1136:{\displaystyle ~u=v=0~}
584:{\displaystyle ~t_{0}~}
2948:
2922:
2896:
2870:
2836:
2789:
2513:
2435:
2371:
2334:
2300:
2205:
2171:
2072:
2037:
1999:
1976:
1950:
1855:
1802:
1747:
1680:
1636:
1610:
1577:
1541:
1434:
1408:
1382:
1356:
1259:
1218:
1173:
1137:
1088:
1058:
1008:
984:In the application in
975:
945:
882:
856:
830:
797:
770:can be interpreted as
764:
684:
654:
585:
552:
505:
475:
441:
381:
321:
187:
157:
131:
3181:10.1007/s002200050164
2949:
2923:
2897:
2871:
2837:
2790:
2514:
2436:
2372:
2335:
2301:
2206:
2172:
2073:
2038:
2000:
1977:
1951:
1856:
1803:
1748:
1681:
1637:
1611:
1578:
1542:
1435:
1409:
1383:
1357:
1260:
1219:
1174:
1138:
1089:
1059:
1009:
976:
946:
883:
857:
831:
798:
765:
685:
655:
586:
553:
506:
476:
442:
382:
322:
188:
158:
132:
79:is a special kind of
3209:Mathematical physics
3095:Journal of Physics A
2932:
2906:
2880:
2854:
2810:
2524:
2448:
2381:
2344:
2312:
2215:
2183:
2082:
2050:
2021:
1989:
1960:
1865:
1812:
1757:
1700:
1646:
1620:
1587:
1554:
1447:
1418:
1392:
1366:
1269:
1231:
1224:is almost periodic.
1187:
1147:
1109:
1072:
1030:
1014:may have a sense of
992:
959:
895:
866:
840:
814:
781:
694:
668:
601:
562:
515:
489:
459:
391:
334:
200:
171:
141:
115:
81:nonlinear oscillator
3173:1997CMaPh.188..251K
3108:2007JPhA...40.2107K
3060:1985ZPhyB..59..111O
3023:1975PhR....18....1T
2947:{\displaystyle ~v~}
2921:{\displaystyle ~u~}
2895:{\displaystyle ~v~}
2869:{\displaystyle ~u~}
2800:significant figures
2308:In the whole range
2013:Period of pulsation
1635:{\displaystyle ~x~}
1487:
1433:{\displaystyle ~z~}
1407:{\displaystyle ~x~}
1381:{\displaystyle ~z~}
1362:does not depend on
1087:{\displaystyle ~x~}
1007:{\displaystyle ~x~}
974:{\displaystyle ~x~}
881:{\displaystyle ~x~}
855:{\displaystyle ~v~}
829:{\displaystyle ~u~}
796:{\displaystyle ~D~}
683:{\displaystyle ~x~}
504:{\displaystyle ~t~}
474:{\displaystyle ~z~}
186:{\displaystyle ~z~}
156:{\displaystyle ~z~}
130:{\displaystyle ~x~}
3068:10.1007/BF01325388
2944:
2918:
2892:
2866:
2846:Decay of pulsation
2832:
2785:
2509:
2431:
2367:
2330:
2296:
2201:
2167:
2068:
2033:
1995:
1972:
1946:
1851:
1798:
1743:
1676:
1632:
1606:
1573:
1537:
1459:
1430:
1404:
1378:
1352:
1255:
1214:
1169:
1133:
1084:
1054:
1004:
971:
953:exponential growth
941:
878:
852:
826:
793:
760:
680:
660:may have sense of
650:
581:
548:
501:
471:
437:
377:
317:
183:
153:
127:
2984:Toda field theory
2982:In contrast, the
2943:
2937:
2917:
2911:
2891:
2885:
2865:
2859:
2765:
2752:
2504:
2491:
2458:
2430:
2426:
2386:
2366:
2349:
2329:
2317:
2295:
2220:
2200:
2188:
2166:
2135:
2087:
2067:
2055:
2032:
2026:
1998:{\displaystyle E}
1971:
1965:
1938:
1911:
1861:; and the energy
1849:
1796:
1742:
1705:
1658:
1631:
1625:
1605:
1592:
1572:
1559:
1535:
1532:
1510:
1429:
1423:
1403:
1397:
1377:
1371:
1351:
1322:
1289:
1274:
1254:
1236:
1213:
1192:
1168:
1152:
1132:
1114:
1083:
1077:
1053:
1035:
1003:
997:
970:
964:
940:
900:
877:
871:
851:
845:
825:
819:
792:
786:
759:
755:
709:
699:
679:
673:
648:
616:
606:
580:
567:
547:
520:
500:
494:
470:
464:
436:
396:
376:
339:
286:
243:
182:
176:
152:
146:
126:
120:
93:solid-state laser
69:
68:
61:
3221:
3193:
3192:
3166:
3144:
3138:
3137:
3119:
3089:
3080:
3079:
3041:
3035:
3034:
3004:
2967:Continuous limit
2953:
2951:
2950:
2945:
2941:
2935:
2927:
2925:
2924:
2919:
2915:
2909:
2901:
2899:
2898:
2893:
2889:
2883:
2875:
2873:
2872:
2867:
2863:
2857:
2841:
2839:
2838:
2833:
2831:
2830:
2794:
2792:
2791:
2786:
2784:
2783:
2779:
2770:
2766:
2764:
2763:
2762:
2753:
2751:
2750:
2738:
2733:
2732:
2717:
2716:
2701:
2700:
2685:
2684:
2669:
2668:
2653:
2652:
2624:
2623:
2622:
2610:
2609:
2594:
2593:
2578:
2577:
2562:
2561:
2533:
2518:
2516:
2515:
2510:
2505:
2503:
2493:
2492:
2489:
2482:
2474:
2460:
2459:
2456:
2440:
2438:
2437:
2432:
2428:
2427:
2425:
2411:
2403:
2384:
2376:
2374:
2373:
2368:
2364:
2363:
2347:
2339:
2337:
2336:
2331:
2327:
2315:
2305:
2303:
2302:
2297:
2293:
2292:
2288:
2281:
2256:
2255:
2251:
2218:
2210:
2208:
2207:
2202:
2198:
2186:
2176:
2174:
2173:
2168:
2164:
2163:
2159:
2155:
2154:
2136:
2131:
2130:
2121:
2085:
2077:
2075:
2074:
2069:
2065:
2053:
2042:
2040:
2039:
2034:
2030:
2024:
2004:
2002:
2001:
1996:
1981:
1979:
1978:
1973:
1969:
1963:
1955:
1953:
1952:
1947:
1939:
1934:
1923:
1912:
1910:
1890:
1860:
1858:
1857:
1852:
1850:
1845:
1828:
1807:
1805:
1804:
1799:
1797:
1792:
1791:
1790:
1778:
1777:
1767:
1752:
1750:
1749:
1744:
1740:
1730:
1729:
1720:
1715:
1714:
1703:
1685:
1683:
1682:
1677:
1660:
1659:
1651:
1641:
1639:
1638:
1633:
1629:
1623:
1615:
1613:
1612:
1607:
1603:
1602:
1601:
1590:
1582:
1580:
1579:
1574:
1570:
1569:
1568:
1557:
1546:
1544:
1543:
1538:
1536:
1534:
1533:
1513:
1511:
1506:
1503:
1499:
1498:
1491:
1486:
1485:
1484:
1474:
1473:
1472:
1439:
1437:
1436:
1431:
1427:
1421:
1413:
1411:
1410:
1405:
1401:
1395:
1387:
1385:
1384:
1379:
1375:
1369:
1361:
1359:
1358:
1353:
1349:
1333:
1332:
1327:
1323:
1321:
1317:
1316:
1309:
1305:
1304:
1297:
1290:
1282:
1272:
1264:
1262:
1261:
1256:
1252:
1234:
1223:
1221:
1220:
1215:
1211:
1190:
1178:
1176:
1175:
1170:
1166:
1165:
1164:
1150:
1142:
1140:
1139:
1134:
1130:
1112:
1093:
1091:
1090:
1085:
1081:
1075:
1063:
1061:
1060:
1055:
1051:
1033:
1013:
1011:
1010:
1005:
1001:
995:
980:
978:
977:
972:
968:
962:
950:
948:
947:
942:
938:
925:
924:
898:
887:
885:
884:
879:
875:
869:
861:
859:
858:
853:
849:
843:
835:
833:
832:
827:
823:
817:
802:
800:
799:
794:
790:
784:
769:
767:
766:
761:
757:
756:
754:
753:
752:
743:
742:
735:
731:
730:
725:
724:
716:
711:
710:
702:
697:
689:
687:
686:
681:
677:
671:
659:
657:
656:
651:
649:
647:
643:
642:
635:
631:
630:
623:
618:
617:
609:
604:
590:
588:
587:
582:
578:
577:
576:
565:
557:
555:
554:
549:
545:
544:
543:
534:
518:
510:
508:
507:
502:
498:
492:
480:
478:
477:
472:
468:
462:
451:Physical meaning
446:
444:
443:
438:
434:
421:
420:
394:
386:
384:
383:
378:
374:
367:
366:
337:
326:
324:
323:
318:
298:
287:
285:
281:
280:
273:
269:
268:
261:
244:
242:
241:
240:
231:
230:
223:
219:
218:
217:
216:
204:
192:
190:
189:
184:
180:
174:
162:
160:
159:
154:
150:
144:
136:
134:
133:
128:
124:
118:
109:dynamical system
97:transient regime
64:
57:
53:
50:
44:
24:
23:
16:
3229:
3228:
3224:
3223:
3222:
3220:
3219:
3218:
3199:
3198:
3197:
3196:
3146:
3145:
3141:
3117:10.1.1.535.5379
3091:
3090:
3083:
3043:
3042:
3038:
3010:Physics Reports
3006:
3005:
2998:
2993:
2969:
2930:
2929:
2904:
2903:
2878:
2877:
2852:
2851:
2848:
2819:
2808:
2807:
2754:
2742:
2724:
2708:
2692:
2676:
2660:
2644:
2625:
2614:
2601:
2585:
2569:
2553:
2534:
2528:
2527:
2522:
2521:
2484:
2483:
2475:
2451:
2446:
2445:
2412:
2404:
2379:
2378:
2342:
2341:
2310:
2309:
2261:
2257:
2239:
2213:
2212:
2181:
2180:
2146:
2122:
2113:
2109:
2080:
2079:
2078:, the period
2048:
2047:
2019:
2018:
2015:
1987:
1986:
1958:
1957:
1924:
1894:
1863:
1862:
1829:
1810:
1809:
1782:
1769:
1768:
1755:
1754:
1721:
1706:
1698:
1697:
1644:
1643:
1618:
1617:
1593:
1585:
1584:
1560:
1552:
1551:
1504:
1492:
1476:
1464:
1445:
1444:
1416:
1415:
1390:
1389:
1364:
1363:
1310:
1298:
1292:
1291:
1267:
1266:
1229:
1228:
1185:
1184:
1153:
1145:
1144:
1107:
1106:
1103:
1070:
1069:
1028:
1027:
990:
989:
957:
956:
916:
893:
892:
864:
863:
838:
837:
812:
811:
779:
778:
744:
736:
718:
717:
692:
691:
666:
665:
636:
624:
599:
598:
568:
560:
559:
535:
513:
512:
487:
486:
457:
456:
453:
412:
389:
388:
358:
332:
331:
291:
274:
262:
232:
224:
208:
205:
198:
197:
169:
168:
139:
138:
113:
112:
105:
77:Toda oscillator
65:
54:
48:
45:
37:help improve it
34:
25:
21:
12:
11:
5:
3227:
3225:
3217:
3216:
3211:
3201:
3200:
3195:
3194:
3164:hep-th/9507065
3157:(2): 251–266.
3139:
3081:
3054:(1): 111–115.
3036:
2995:
2994:
2992:
2989:
2968:
2965:
2940:
2914:
2888:
2862:
2847:
2844:
2829:
2826:
2822:
2818:
2815:
2804:relative error
2798:to at least 8
2796:
2795:
2782:
2778:
2774:
2769:
2761:
2757:
2749:
2745:
2741:
2736:
2731:
2727:
2723:
2720:
2715:
2711:
2707:
2704:
2699:
2695:
2691:
2688:
2683:
2679:
2675:
2672:
2667:
2663:
2659:
2656:
2651:
2647:
2643:
2640:
2637:
2634:
2631:
2628:
2621:
2617:
2613:
2608:
2604:
2600:
2597:
2592:
2588:
2584:
2581:
2576:
2572:
2568:
2565:
2560:
2556:
2552:
2549:
2546:
2543:
2540:
2537:
2531:
2519:
2508:
2502:
2499:
2496:
2487:
2481:
2478:
2472:
2469:
2466:
2463:
2454:
2424:
2421:
2418:
2415:
2410:
2407:
2401:
2398:
2395:
2392:
2389:
2377:and frequency
2362:
2359:
2356:
2353:
2326:
2323:
2320:
2291:
2287:
2284:
2280:
2276:
2273:
2270:
2267:
2264:
2260:
2254:
2250:
2246:
2242:
2238:
2235:
2232:
2229:
2226:
2223:
2211:, the period
2197:
2194:
2191:
2162:
2158:
2153:
2149:
2145:
2142:
2139:
2134:
2129:
2125:
2119:
2116:
2112:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2064:
2061:
2058:
2029:
2014:
2011:
1994:
1968:
1945:
1942:
1937:
1933:
1930:
1927:
1921:
1918:
1915:
1909:
1906:
1903:
1900:
1897:
1893:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1848:
1844:
1841:
1838:
1835:
1832:
1826:
1823:
1820:
1817:
1795:
1789:
1785:
1781:
1776:
1772:
1765:
1762:
1739:
1736:
1733:
1728:
1724:
1719:
1713:
1709:
1675:
1672:
1669:
1666:
1663:
1657:
1654:
1628:
1600:
1596:
1567:
1563:
1548:
1547:
1531:
1528:
1525:
1522:
1519:
1516:
1509:
1502:
1497:
1483:
1479:
1471:
1467:
1462:
1458:
1455:
1452:
1426:
1400:
1374:
1348:
1345:
1342:
1339:
1336:
1331:
1326:
1320:
1315:
1308:
1303:
1295:
1288:
1285:
1280:
1277:
1251:
1248:
1245:
1242:
1239:
1210:
1207:
1204:
1201:
1198:
1195:
1163:
1160:
1156:
1129:
1126:
1123:
1120:
1117:
1102:
1099:
1080:
1050:
1047:
1044:
1041:
1038:
1000:
967:
937:
934:
931:
928:
923:
919:
915:
912:
909:
906:
903:
891:The potential
874:
848:
822:
789:
751:
747:
741:
734:
729:
723:
714:
708:
705:
676:
646:
641:
634:
629:
621:
615:
612:
575:
571:
542:
538:
533:
529:
526:
523:
497:
467:
452:
449:
433:
430:
427:
424:
419:
415:
411:
408:
405:
402:
399:
373:
370:
365:
361:
357:
354:
351:
348:
345:
342:
328:
327:
316:
313:
310:
307:
304:
301:
297:
294:
290:
284:
279:
272:
267:
259:
256:
253:
250:
247:
239:
235:
229:
222:
215:
211:
179:
149:
123:
104:
101:
89:self-pulsation
67:
66:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
3226:
3215:
3212:
3210:
3207:
3206:
3204:
3190:
3186:
3182:
3178:
3174:
3170:
3165:
3160:
3156:
3152:
3151:
3143:
3140:
3135:
3131:
3127:
3123:
3118:
3113:
3109:
3105:
3101:
3097:
3096:
3088:
3086:
3082:
3077:
3073:
3069:
3065:
3061:
3057:
3053:
3049:
3048:
3040:
3037:
3032:
3028:
3024:
3020:
3016:
3012:
3011:
3003:
3001:
2997:
2990:
2988:
2985:
2980:
2978:
2974:
2966:
2964:
2962:
2961:optical bench
2959:laser at the
2958:
2938:
2912:
2886:
2860:
2845:
2843:
2827:
2824:
2820:
2816:
2813:
2805:
2801:
2780:
2776:
2772:
2767:
2759:
2755:
2747:
2743:
2739:
2734:
2729:
2725:
2721:
2718:
2713:
2709:
2705:
2702:
2697:
2693:
2689:
2686:
2681:
2677:
2673:
2670:
2665:
2661:
2657:
2654:
2649:
2645:
2641:
2638:
2635:
2632:
2629:
2626:
2619:
2615:
2611:
2606:
2602:
2598:
2595:
2590:
2586:
2582:
2579:
2574:
2570:
2566:
2563:
2558:
2554:
2550:
2547:
2544:
2541:
2538:
2535:
2529:
2520:
2506:
2497:
2485:
2479:
2476:
2470:
2464:
2452:
2444:
2443:
2442:
2419:
2413:
2408:
2405:
2399:
2393:
2387:
2357:
2351:
2340:, the period
2324:
2321:
2318:
2306:
2289:
2282:
2278:
2274:
2268:
2265:
2262:
2258:
2252:
2248:
2244:
2240:
2236:
2233:
2227:
2221:
2195:
2192:
2189:
2177:
2160:
2151:
2147:
2140:
2137:
2132:
2127:
2123:
2117:
2114:
2110:
2106:
2103:
2100:
2094:
2088:
2062:
2059:
2056:
2044:
2027:
2012:
2010:
2008:
1992:
1983:
1966:
1943:
1940:
1935:
1931:
1928:
1925:
1919:
1916:
1913:
1904:
1898:
1895:
1891:
1886:
1880:
1874:
1871:
1868:
1846:
1839:
1833:
1830:
1824:
1821:
1818:
1815:
1793:
1783:
1779:
1770:
1763:
1760:
1737:
1734:
1731:
1722:
1717:
1707:
1694:
1692:
1687:
1673:
1670:
1664:
1655:
1652:
1626:
1594:
1561:
1526:
1517:
1514:
1507:
1500:
1477:
1465:
1460:
1456:
1453:
1450:
1443:
1442:
1441:
1424:
1398:
1372:
1343:
1334:
1329:
1324:
1318:
1306:
1293:
1286:
1283:
1278:
1275:
1249:
1246:
1243:
1240:
1237:
1225:
1205:
1199:
1196:
1193:
1182:
1161:
1158:
1154:
1127:
1124:
1121:
1118:
1115:
1100:
1098:
1095:
1078:
1067:
1045:
1039:
1036:
1025:
1021:
1017:
998:
987:
986:laser physics
982:
965:
954:
935:
932:
929:
926:
921:
917:
913:
907:
889:
872:
846:
820:
808:
806:
787:
775:
773:
749:
745:
732:
727:
712:
706:
703:
674:
663:
644:
632:
619:
613:
610:
597:
592:
591:is constant.
573:
569:
540:
536:
531:
527:
524:
521:
495:
484:
481:has sense of
465:
450:
448:
431:
428:
425:
422:
417:
413:
409:
403:
371:
368:
363:
359:
355:
352:
346:
340:
314:
311:
308:
302:
295:
288:
282:
270:
254:
248:
245:
237:
233:
220:
213:
196:
195:
194:
177:
166:
147:
121:
110:
102:
100:
98:
94:
90:
86:
85:Morikazu Toda
82:
78:
74:
63:
60:
52:
42:
38:
32:
29:This article
27:
18:
17:
3154:
3148:
3142:
3099:
3093:
3051:
3045:
3039:
3014:
3008:
2981:
2970:
2957:self-pulsing
2849:
2797:
2307:
2178:
2045:
2016:
1984:
1695:
1688:
1549:
1227:In the case
1226:
1104:
1096:
1024:output power
1020:laser cavity
983:
890:
809:
776:
772:acceleration
593:
454:
329:
106:
76:
70:
55:
46:
30:
3102:(9): 1–18.
2007:quasienergy
1066:oscillation
3203:Categories
2991:References
2973:Toda chain
1696:The ratio
596:derivative
103:Definition
3112:CiteSeerX
3076:119657810
2825:−
2817:×
2756:γ
2744:π
2726:γ
2710:γ
2694:γ
2678:γ
2662:γ
2646:γ
2636:γ
2616:γ
2603:γ
2587:γ
2571:γ
2555:γ
2545:γ
2498:γ
2480:π
2465:γ
2420:γ
2409:π
2394:γ
2358:γ
2319:γ
2283:γ
2241:γ
2228:γ
2193:≫
2190:γ
2148:γ
2124:γ
2107:π
2095:γ
2060:≪
2057:γ
2028:γ
1967:γ
1941:−
1936:γ
1932:γ
1929:
1920:
1905:γ
1899:
1892:γ
1881:γ
1847:γ
1840:γ
1834:
1825:
1816:δ
1780:−
1761:δ
1738:γ
1656:˙
1521:Φ
1518:−
1461:∫
1457:±
1338:Φ
1159:−
1040:
1016:logarithm
933:−
927:−
902:Φ
707:¨
614:˙
429:−
423:−
398:Φ
293:Φ
165:evolution
49:July 2012
3189:17196702
3134:53330023
3017:(1): 1.
2674:164.6842
2658:303.2428
2599:4.082607
2583:16.86221
2567:191.4489
2551:695.2419
805:friction
662:velocity
558:, where
296:′
3169:Bibcode
3104:Bibcode
3056:Bibcode
3019:Bibcode
2802:. The
2690:36.6434
690:; then
330:where
95:in the
73:physics
35:Please
3187:
3132:
3114:
3074:
2942:
2936:
2916:
2910:
2890:
2884:
2864:
2858:
2722:0.8983
2706:3.9596
2429:
2385:
2365:
2348:
2328:
2316:
2294:
2219:
2199:
2187:
2165:
2086:
2066:
2054:
2031:
2025:
1970:
1964:
1741:
1704:
1630:
1624:
1604:
1591:
1571:
1558:
1550:where
1428:
1422:
1402:
1396:
1376:
1370:
1350:
1273:
1253:
1235:
1212:
1191:
1181:period
1167:
1151:
1131:
1113:
1101:Energy
1082:
1076:
1052:
1034:
1002:
996:
969:
963:
939:
899:
876:
870:
850:
844:
824:
818:
791:
785:
758:
698:
678:
672:
605:
579:
566:
546:
519:
499:
493:
469:
463:
435:
395:
375:
338:
181:
175:
151:
145:
125:
119:
75:, the
3185:S2CID
3159:arXiv
3130:S2CID
3072:S2CID
2627:10630
2536:10630
2179:When
2046:When
2971:The
2928:and
2876:and
2642:2467
2322:>
1926:sinh
1896:tanh
1583:and
1414:and
836:and
594:The
483:time
3177:doi
3155:188
3122:doi
3064:doi
3027:doi
2633:674
2542:674
2490:fit
2457:fit
1831:sin
1808:as
1788:min
1775:max
1727:min
1712:max
1599:max
1566:min
1482:max
1470:min
1068:of
1037:exp
71:In
39:to
3205::
3183:.
3175:.
3167:.
3153:.
3128:.
3120:.
3110:.
3100:40
3098:.
3084:^
3070:.
3062:.
3052:59
3050:.
3025:.
3015:18
3013:.
2999:^
2842:.
2821:10
2814:22
2740:16
2133:24
2043:.
2009:.
1982:.
1917:ln
1822:ln
1693:.
1686:.
1155:10
1094:.
988:,
981:.
888:.
807:.
387:,
99:.
3191:.
3179::
3171::
3161::
3136:.
3124::
3106::
3078:.
3066::
3058::
3033:.
3029::
3021::
2939:v
2913:u
2887:v
2861:u
2828:9
2781:4
2777:/
2773:1
2768:)
2760:8
2748:4
2735:+
2730:7
2719:+
2714:6
2703:+
2698:5
2687:+
2682:4
2671:+
2666:3
2655:+
2650:2
2639:+
2630:+
2620:6
2612:+
2607:5
2596:+
2591:4
2580:+
2575:3
2564:+
2559:2
2548:+
2539:+
2530:(
2507:=
2501:)
2495:(
2486:T
2477:2
2471:=
2468:)
2462:(
2453:k
2423:)
2417:(
2414:T
2406:2
2400:=
2397:)
2391:(
2388:k
2361:)
2355:(
2352:T
2325:0
2290:)
2286:)
2279:/
2275:1
2272:(
2269:O
2266:+
2263:1
2259:(
2253:2
2249:/
2245:1
2237:4
2234:=
2231:)
2225:(
2222:T
2196:1
2161:)
2157:)
2152:4
2144:(
2141:O
2138:+
2128:2
2118:+
2115:1
2111:(
2104:2
2101:=
2098:)
2092:(
2089:T
2063:1
1993:E
1944:1
1914:+
1908:)
1902:(
1887:=
1884:)
1878:(
1875:E
1872:=
1869:E
1843:)
1837:(
1819:=
1794:1
1784:x
1771:x
1764:=
1735:2
1732:=
1723:x
1718:/
1708:x
1674:0
1671:=
1668:)
1665:0
1662:(
1653:x
1627:x
1595:x
1562:x
1530:)
1527:a
1524:(
1515:E
1508:2
1501:a
1496:d
1478:x
1466:x
1454:=
1451:z
1425:z
1399:x
1373:z
1347:)
1344:x
1341:(
1335:+
1330:2
1325:)
1319:z
1314:d
1307:x
1302:d
1294:(
1287:2
1284:1
1279:=
1276:E
1250:0
1247:=
1244:v
1241:=
1238:u
1209:)
1206:t
1203:(
1200:x
1197:=
1194:x
1162:4
1128:0
1125:=
1122:v
1119:=
1116:u
1079:x
1049:)
1046:x
1043:(
999:x
966:x
936:1
930:x
922:x
918:e
914:=
911:)
908:x
905:(
873:x
847:v
821:u
788:D
750:2
746:z
740:d
733:x
728:2
722:d
713:=
704:x
675:x
645:z
640:d
633:x
628:d
620:=
611:x
574:0
570:t
541:0
537:t
532:/
528:t
525:=
522:z
496:t
466:z
432:1
426:x
418:x
414:e
410:=
407:)
404:x
401:(
372:v
369:+
364:x
360:e
356:u
353:=
350:)
347:x
344:(
341:D
315:,
312:0
309:=
306:)
303:x
300:(
289:+
283:z
278:d
271:x
266:d
258:)
255:x
252:(
249:D
246:+
238:2
234:z
228:d
221:x
214:2
210:d
178:z
148:z
122:x
62:)
56:(
51:)
47:(
33:.
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