×

Long-time asymptotic behavior for the discrete defocusing mKdV equation. (English) Zbl 1434.35152

Summary: In this article, we apply Deift-Zhou nonlinear steepest descent method to analyze the long-time asymptotic behavior of the solution for the discrete defocusing mKdV equation \[\dot{q}_n = \left( 1-q_n^2\right) (q_{n+1}-q_{n-1}) \] with decay initial value \[ q_n(t=0) = q_n(0), \] where \(n=0,\pm 1,\pm 2,\dots\) is a discrete variable and \(t\) is continuous time variable. This equation was proposed by Ablowitz and Ladik.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
35Q15 Riemann-Hilbert problems in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ablowitz, MJ, Nonlinear evolution equations continuous and discrete, SIAM Rev., 19, 4, 663-684 (1977) · Zbl 0375.35030 · doi:10.1137/1019105
[2] Beals, R.; Coifman, RR, Scattering and inverse scattering for first order systems, Commun. Pure Appl. Math., 37, 1, 39-90 (1984) · Zbl 0514.34021 · doi:10.1002/cpa.3160370105
[3] Cheng, P.; Venakides, S.; Zhou, X., Long-time asymptotics for the pure radiation solution of the sine-gordon equation, Hist. Philos. Logic., 24, 7-8, 1195-1262 (1999) · Zbl 0937.35154
[4] De Monvel, AB; Kostenko, A.; Shepelsky, D.; Teschl, G., Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41, 4, 1559-1588 (2009) · Zbl 1204.37073 · doi:10.1137/090748500
[5] Deift, P.A., Its, A.R., Zhou, X.: Long-time asymptotics for integrable nonlinear wave equations. In: Fokas, A.S., Zakharov, V.E. (eds.) Important Developments in Soliton Theory, pp. 181-204. Springer (1993) · Zbl 0926.35132
[6] Deift, PA; Zhou, X., Long-Time Behavior of the Non-focusing Nonlinear Schrödinger Equation—A Case Study (1994), Tokyo: University of Tokyo, Tokyo
[7] Deift, PA; Zhou, X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. Math., 137, 2, 295-368 (2017) · Zbl 0771.35042 · doi:10.2307/2946540
[8] Grunert, K.; Teschl, G., Long-time asymptotics for the Korteweg-de Vries equation via nonlinear steepest descent, Math. Phys. Anal. Geom., 12, 3, 287-324 (2009) · Zbl 1179.37098 · doi:10.1007/s11040-009-9062-2
[9] Krüger, H.; Teschl, G., Long-time asymptotics of the Toda lattice for decaying initial data revisited, Rev. Math. Phys., 21, 1, 61-109 (2009) · Zbl 1173.37057 · doi:10.1142/S0129055X0900358X
[10] Lang, S., Differential and Riemannian Manifolds (2012), Berlin: Springer, Berlin
[11] Narita, K., Miura transformations between Sokolov-Shabat’s equation and the discrete MKdV equation, J. Phys. Soc. Jpn., 66, 12, 4047-4048 (1997) · doi:10.1143/JPSJ.66.4047
[12] Vartanian, AH, Higher order asymptotics of the modified non-linear Schrödinger equation, Commun. Partial Differ. Equ., 25, 5-6, 1043-1098 (2000) · Zbl 0952.35126 · doi:10.1080/03605300008821541
[13] Wang, Z.; Zou, L.; Zhang, HQ, Solitary solution of discrete mKdV equation by homotopy analysis method, Commun. Theor. Phys., 49, 6, 1373 (2008) · Zbl 1392.34080 · doi:10.1088/0253-6102/49/6/03
[14] Wen, XY; Gao, YT, Darboux transformation and explicit solutions for discretized modified Korteweg-de Vries lattice equation, Commun. Theor. Phys., 53, 5, 825 (2010) · Zbl 1219.35266 · doi:10.1088/0253-6102/53/5/07
[15] Yamane, H., Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation, J. Math. Soc. Jpn., 66, 3, 765-803 (2014) · Zbl 1309.35147 · doi:10.2969/jmsj/06630765
[16] Yamane, H., Long-time asymptotics for the defocusing integrable discrete nonlinear Schrodinger equation II, Symm. Integr. Geom. Methods Appl., 11, 020 (2015) · Zbl 1311.35301
[17] Yamane, H., Riemann-Hilbert factorization of matrices invariant under inversion in a circle, Proc. Am. Math. Soc., 147, 2147-2157 (2019) · Zbl 1458.35296 · doi:10.1090/proc/14398
[18] Yamane, H., Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation, Funkcialaj Ekvacioj-Serio Inteenacia, 62, 227-253 (2019) · Zbl 1434.35197 · doi:10.1619/fesi.62.227
[19] Zhu, SD, Exp-function method for the discrete mKdV lattice, Int. J. Nonlinear Sci. Numer. Simul., 8, 3, 465-468 (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.