On long-time asymptotics to the nonlocal short pulse equation with the Schwartz-type initial data: without solitons. (English) Zbl 1514.35057
Summary: In this work, the initial value problem of the nonlocal short pulse (NSP) equation is studied with the Schwartz-type initial data. Our aim is to adequately study the long-time asymptotic behavior of the solution of the NSP equation in view of the initial value condition. Starting with the Lax pair of the NSP equation, we give some discussion about the NSP equation, including infinite number of conservation laws and finite-dimensional Hamiltonian function. The relevant spectral analysis is discussed, among which it is important to note that we discuss it separately due to the singularity of the spectrum. According to these results, a suitable Riemann-Hilbert (RH) problem which is used to express the solution of the NSP equation is established. By using the nonlinear steepest descent method, the original RH problem is transformed into a model RH problem, which is given by a parabolic cylinder function. The long-time asymptotic solution of the NSP equation is therefore given, especially the error term is quite different from the local situation.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35R09 | Integro-partial differential equations |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
Keywords:
nonlocal short pulse equation; zero boundary condition; Riemann-Hilbert problem; nonlinear steepest descent method; long-time asymptoticsSoftware:
PSEUDOReferences:
| [1] | Ablowitz, M. J.; Feng, B. F.; Luo, X. D.; Musslimani, Z. H., Inverse scattering transform for the nonlocal reverse space-time sine-gordon, sinh-gordon and nonlinear Schrödinger equations with nonzero boundary conditions, Theoret. Math. Phys., 196, 1241-1267 (2018) · Zbl 1408.35169 |
| [2] | Ablowitz, M. J.; Musslimani, Z. H., Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett., 110, Article 064105 pp. (2013) |
| [3] | Ji, J.; Huang, Z.; Zhu, Z., Reverse space and time nonlocal coupled dispersionless equation and its solutions, Ann. Math. Sci. Appl., 2, 409-429 (2017) · Zbl 1384.35120 |
| [4] | Gürses, M.; Pekcan, A., Nonlocal modified kdv equations and their soliton solutions, Commun. Nonlinear Sci. Numer. Simul., 67, 427-448 (2019) · Zbl 1508.35121 |
| [5] | Priya, N. V.; Senthilvelan, M.; Rangarajan, G.; Lakshmanan, M., On symmetry preserving and symmetry broken bright, dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation, Phys. Lett. A, 383, 15-26 (2019) · Zbl 1404.35422 |
| [6] | Stalin, S.; Senthilvelan, M.; Lakshmanan, M., Degenerate soliton solutions and their dynamics in the nonlocal manakov system: i symmetry preserving and symmetry breaking solutions, Nonlinear Dynam., 95, 343-360 (2019) · Zbl 1439.35447 |
| [7] | Vinayagam, P. S.; Radha, R.; Al Khawaja, U.; Ling, L., Collisional dynamics of solitons in the coupled PT symmetric nonlocal nonlinear Schrödinger equations, Commun. Nonlinear Sci. Numer. Simul., 52, 1-10 (2017) · Zbl 1524.37069 |
| [8] | Ablowitz, M. J.; Musslimani, Z. H., Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation, Nonlinearity, 29, 915 (2016) · Zbl 1338.37099 |
| [9] | Yang, B.; Yang, J., Transformations between nonlocal and local integrable equations, Stud. Appl. Math., 140, 2, 178-201 (2018) · Zbl 1392.35297 |
| [10] | Schäfer, T.; Wayne, C. E., Propagation of ultra-short optical pulses in cubic nonlinear media, Physica D, 196, 90-105 (2004) · Zbl 1054.81554 |
| [11] | Chung, Y.; Jones, C. K.R. T.; Schäfer, T.; Wayne, C. E., Ultra-short pulses in linear and nonlinear media, Nonlinearity, 18, 1351-1374 (2005) · Zbl 1125.35412 |
| [12] | Brunelli, J. C., The short pulse equation hierarchy, J. Math. Phys., 46, Article 123507 pp. (2005) · Zbl 1111.35056 |
| [13] | Brunelli, J. C., The bi-Hamiltonian structure of the short pulse equation, Phys. Lett. A, 353, 475-478 (2006) · Zbl 1181.37094 |
| [14] | Sakovich, A.; Sakovich, S., Solitary wave solutions of the short pulse equation, J. Phys. A: Math. Gen., 39, L361-L367 (2006) · Zbl 1092.81531 |
| [15] | Matsuno, Y., Multiloop solutions and multibreather solutions of the short pulse model equation, J. Phys. Soc. Japan, 76, Article 084003 pp. (2007) |
| [16] | Coclite, G. M.; di Ruvo, L., Well-posedness results for the short pulse equation, Z. Angew. Math. Phys., 66, 4, 1529-1557 (2015) · Zbl 1327.35070 |
| [17] | Pelinovsky, D.; Sakovich, A., Global well-posedness of the short-pulse and sine-Gordon equations in energy space, Commun. Partial Differ. Equ., 35, 4, 613-629 (2010) · Zbl 1204.35010 |
| [18] | Liu, Y.; Pelinovsky, D.; Sakovich, A., Wave breaking in the short-pulse equation, Dyn. Partial Differ. Equ., 6, 4, 291-310 (2009) · Zbl 1190.35061 |
| [19] | Sakovich, S., Integrability of the vector short pulse equation, J. Phys. Soc. Japan, 77, Article 123001 pp. (2008) |
| [20] | Feng, B. F., Complex short pulse and couple complex short pulse equations, Physica D, 297, 62-75 (2015) · Zbl 1392.35069 |
| [21] | Feng, B. F.; Maruno, K.; Ohta, Y., Integrable discretizations of the short pulse equation, J. Phys. A: Math. Gen., 43, Article 085203 pp. (2010) · Zbl 1189.78051 |
| [22] | Zakharov, V. E.; Manakov, S. V., Asymptotic behavior of nonlinear wave systems integrated by the inverse scattering method, Sov. Phys.—JETP, 44, 106-112 (1976) |
| [23] | Deift, P.; Zhou, X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math., 137, 2, 295-368 (1993) · Zbl 0771.35042 |
| [24] | Deift, P.; Its, A. R.; Zhou, X., Long-time asymptotics for integrable nonlinear wave equations, Important developments in soliton theory, Nonlinear Dynam., 181-204 (1993) · Zbl 0926.35132 |
| [25] | Biondini, G.; Mantzavinos, D., Long-time asymptotics for the focusing nonlinear Schrödinger equation with nonzero boundary conditions at infinity and asymptotic stage of modulational instability, Comm. Pure Appl. Math., 70, 12, 2300-2365 (2017) · Zbl 1379.35286 |
| [26] | Xu, J.; Fan, E. G.; Chen, Y., Long-time asymptotic for the derivative nonlinear Schrödinger equation with step-like initial value, Math. Phys. Anal. Geom., 16, 253-288 (2013) · Zbl 1274.35364 |
| [27] | Grunert, K.; Teschl, G., Long-time asymptotics for the Korteweg-de Vries equation via nonlinear steepest descent, Math. Phys. Anal. Geom., 12, 287 (2009) · Zbl 1179.37098 |
| [28] | Liu, N.; Guo, B., Long-time asymptotics for the initial-boundary value problem of coupled Hirota equation on the half-line, Sci. China Math., 64, 1 (2020) |
| [29] | Kitaev, A. V.; Vartanian, A. H., Leading-order temporal asymptotics of the modified nonlinear Schrödinger equation: solitonless sector, Inverse Problems, 13, 1311-1339 (1997) · Zbl 0883.35107 |
| [30] | Kitaev, A. V.; Vartanian, A. H., Asymptotics of solutions to the modified nonlinear Schrödinger equation: solution on a nonvanishing continuous background, SIAM J. Math. Anal., 30, 787-832 (1999) · Zbl 0958.35127 |
| [31] | Xu, J.; Fan, E. G., Long-time asymptotics for the Fokas-Lenells equation with decaying initial value problem: Without solitons, J. Differ. Equ., 259, 1098-1148 (2015) · Zbl 1317.35169 |
| [32] | Boutet de Monvel, A.; Kostenko, A.; Shepelsky, D.; Teschl, G., Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41, 1559-1588 (2009) · Zbl 1204.37073 |
| [33] | Geng, X.; Wang, K.; Chen, M., Long-time asymptotics for the spin-1 gross-pitaevskii equation, Comm. Math. Phys., 1-27 (2021) |
| [34] | Xu, J., Long-time asymptotics for the short pulse equation, J. Differ. Equ., 265, 8, 3494-3532 (2018) · Zbl 1394.35308 |
| [35] | Xu, J.; Fan, E. G., Long-time asymptotic behavior for the complex short pulse equation, J. Differ. Equ., 269, 11, 10322-10349 (2020) · Zbl 1447.35066 |
| [36] | Guo, B.; Liu, N., Long-time asymptotics for the Kundu-Eckhaus equation on the half-line, J. Math. Phys., 59, Article 061505 pp. (2018) · Zbl 1398.37061 |
| [37] | Wang, D. S.; Guo, B.; Wang, X., Long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditions, J. Differ. Equ. (2018) |
| [38] | Cheng, P. J.; Venakides, S.; Zhou, X., Long-time asymptotics for the pure radiation solution of the sine-Gordon equation, Commun. Partial Differ. Equ., 24, 1195-1262 (1999) · Zbl 0937.35154 |
| [39] | Chen, S.; Yan, Z., Long-time asymptotics of solutions for the coupled dispersive AB system with initial value problems, J. Math. Anal. Appl., Article 124401 pp. (2020) |
| [40] | Lenells, J.; Fokas, A. S., On a novel integrable generalization of the sine-Gordon equation, J. Math. Phys., 51, Article 023519 pp. (2010) · Zbl 1309.35138 |
| [41] | Zhou, R. G., Nonlinearization of spectral problems of the nonlinear Schrödinger equation and the real-valued modified Korteweg-de Vries equation, J. Math. Phys., 48, Article 013510 pp. (2007) · Zbl 1121.35127 |
| [42] | Zhou, R. G., Finite-dimensional integrable Hamiltonian systems related to the nonlinear Schrödinger equation, Stud. Appl. Math., 123, 311-335 (2009) · Zbl 1181.35046 |
| [43] | Fokas, A., A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 453, 1411-1443 (1997) · Zbl 0876.35102 |
| [44] | Fokas, A., A Unified Approach to Boundary Value Problems (2008), SIAM: SIAM Philadelphia · Zbl 1181.35002 |
| [45] | Borghese, M.; Jenkins, R.; Mclaughlin, T. R., Long time asymptotic behavior of the focusing nonlinear Schrödinger equation, Ann. Inst. H. Poincaré - AN, 35, 887-920 (2018) · Zbl 1390.35020 |
| [46] | Xu, J., On zeros of the scattering data of the initial value problem for the short pulse equation, Appl. Math. Lett., 94, 8-12 (2019) · Zbl 1417.35082 |
| [47] | Rybalko, Y.; Shepelsky, D., Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation, J. Math. Phys., 60, Article 031504 pp. (2019) · Zbl 1436.35293 |
| [48] | Rybalko, Y.; Shepelsky, D., Long-time asymptotics for the nonlocal nonlinear Schrödinger equation with step-like initial data, J. Differ. Equ., 270, 694-724 (2021) · Zbl 1451.35194 |
| [49] | Lenells, J., The nonlinear steepest descent method for Riemann-Hilbert problems of low regularity, Indiana Univ. Math. J., 66, 1287-1332 (2017) · Zbl 1377.41020 |
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