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The massive Thirring system in the quarter plane

Part of: Infinite-dimensional Hamiltonian systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  23 April 2019

Baoqiang Xia
Baoqiang Xia*
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu221116, P. R. China (xiabaoqiang@126.com)
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Abstract

The unified transform method (UTM) or Fokas method for analyzing initial-boundary value (IBV) problems provides an important generalization of the inverse scattering transform (IST) method for analyzing initial value problems. In comparison with the IST, a major difficulty of the implementation of the UTM, in general, is the involvement of unknown boundary values. In this paper we analyze the IBV problem for the massive Thirring model in the quarter plane, assuming that the initial and boundary data belong to the Schwartz class. We show that for this integrable model, the UTM is as effective as the IST method: Riemann-Hilbert problems we formulated for such a problem have explicit (x, t)-dependence and depend only on the given initial and boundary values; they do not involve additional unknown boundary values.

Keywords

massive Thirring systeminitial-boundary value problemunified transform methodinverse scattering transform

MSC classification

Primary: 35Q51: Soliton-like equations
Secondary: 37K15: Integration of completely integrable systems by inverse spectral and scattering methods
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2387 - 2416
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Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Ablowitz, M. J. and Musslimani, Z. H.. Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110 (2013), 064105.CrossRefGoogle ScholarPubMed
2Ablowitz, M. J. and Musslimani, Z. H.. Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity 29 (2016), 915–946.CrossRefGoogle Scholar
3Ablowitz, M. J. and Musslimani, Z. H.. Integrable nonlocal nonlinear equations. Stud. Appl. Math. 139 (2017), 7–59.Google Scholar
4Biondini, G. and Bui, A.. The Ablowitz-Ladik system with linearizable boundary conditions. J. Phys. A: Math. Theor. 48 (2015), 375202.CrossRefGoogle Scholar
5Biondini, G. and Hwang, G.. Initial-boundary-value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations. Inverse Probl. 24 (2008), 065011.CrossRefGoogle Scholar
6Deift, P. and Zhou, X.. A steepest descent method for oscillatory Riemann-Hilbert problems. Ann. Math. 137 (1993), 295–368.CrossRefGoogle Scholar
7Enolskii, V. Z., Gesztesy, F. and Holden, H.. Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999), 163-200, CMS Conf. Proc., 28, Amer. Math. Soc., Providence, RI, (2000).Google Scholar
8Fokas, A. S.. A unified transform method for solving linear and certain nonlinear PDEs. Proc. R. Soc. London, Ser. A 53 (1997), 1411.CrossRefGoogle Scholar
9Fokas, A. S.. Integrable nonlinear evolution equations on the half-line. Commun. Math. Phys. 230 (2002), 1–39.CrossRefGoogle Scholar
10Fokas, A. S.. The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs. Commun. Pure Appl. Math. 58 (2005), 639–670.CrossRefGoogle Scholar
11Fokas, A. S.. A Unified Approach to Boundary Value Problems,vol. 27 (Philadelphia: Society for Industrial and Applied Mathematics, 2008).CrossRefGoogle Scholar
12Fokas, A. S. and Its, A. R.. The linearization of the initial-boundary value problem of the nonlinear Schrödinger equation. SIAM J. Math. Anal. 27 (1996), 738–764.CrossRefGoogle Scholar
13Fokas, A. S. and Lenells, J.. The Unified Method: I Non-Linearizable Problems on the Half-Line. J. Phys. A: Math. Theor. 45 (2012), 195201.CrossRefGoogle Scholar
14Fokas, A. S., Its, A. R. and Sung, L. Y.. The nonlinear Schrödinger equation on the half-line. Nonlinearity 18 (2005), 1771–1822.CrossRefGoogle Scholar
15Geng, X., Liu, H. and Zhu, J.. Initial-boundary value problems for the coupled nonlinear Schrödinger equation on the half-line. Stud. Appl. Math. 135 (2015), 310–346.CrossRefGoogle Scholar
16Joshi, N. and Pelinovsky, D. E.. Integrable semi-discretization of the massive Thirring system in laboratory coordinates. J. Phys. A: Math. Theor. 52 (2019), 03LT01 (12pp).CrossRefGoogle Scholar
17Kaup, D. J. and Lakoba, T. I.. The squared eigenfunctions of the massive Thirring model in laboratory coordinates. J. Math. Phys. 37 (1996), 308–323.CrossRefGoogle Scholar
18Kaup, D. J. and Newell, A. C.. On the Coleman correspondence and the solution of the Massive Thirring model. Lett. Nuovo Cimento 20 (1977), 325–331.CrossRefGoogle Scholar
19Kawata, T., Morishima, T. and Inoue, H.. Inverse scattering method for the two-dimensional massive Thirring model. J. Phys. Soc. Japan 47 (1979), 1327–1334.CrossRefGoogle Scholar
20Kuznetzov, E. A. and Mikhailov, A. V.. On the complete integrability of the two-dimensional classical Thirring model. Theor. Math. Phys. 30 (1977), 193–200.CrossRefGoogle Scholar
21Lee, J. H.. Solvability of the derivative nonlinear Schrödinger equation and the massive Thirring model. Theoret. Math. Phys. 99 (1994), 617–621.CrossRefGoogle Scholar
22Lenells, J.. Initial-boundary value problems for integrable evolution equations with 3 × 3 Lax pairs. Physica D 241 (2012), 857–875.CrossRefGoogle Scholar
23Lenells, J.. The Degasperis-Procesi equation on the half-line. Nonlinear Anal. 76 (2013), 122–139.CrossRefGoogle Scholar
24Martinez Alonso, L.. Soliton classical dynamics in the sine-Gordon equation in terms of the massive Thirring model. Phys. Rev. D 30 (1984), 2595–2601.CrossRefGoogle Scholar
25Mikhailov, A. V.. Integrability of the two-dimensional Thirring model. JETP Lett. 23 (1976), 320–323.Google Scholar
26Monvel, A. B., Fokas, A. S. and Shepelsky, D.. Integrable nonlinear evolution equations on a finite interval. Commun. Math. Phys. 263 (2006), 133–172.CrossRefGoogle Scholar
27Nijhoff, F. W., Capel, H. W. and Quispel, G. R. W.. Integrable lattice version of the massive Thirring model and its linearization. Phys. Lett. A 98 (1983), 83–86.CrossRefGoogle Scholar
28Nijhoff, F. W., Capel, H. W., Quispel, G. R. W. and van der Linden, J.. The derivative nonlinear Schrödinger equation and the massive Thirring model. Phys. Lett. A 93 (1983), 455–458.CrossRefGoogle Scholar
29Orfanidis, S. J.. Soliton solutions of the massive Thirring model and the inverse scattering transform. Phys. Rev. D 14 (1976), 472–478.CrossRefGoogle Scholar
30Pelinovsky, D. E. and Saalmann, A.. Inverse scattering for the massive Thirring model. arXiv: 1801.00039.Google Scholar
31Prikarpatskii, A. K.. Geometrical structure and Bäcklund transformations of nonlinear evolution equations possessing a Lax representation. Theoret. Math. Phys. 46 (1981), 249–256.CrossRefGoogle Scholar
32Saalmann, A.. Long-time asymptotics for the massive Thirring model. arXiv: 1807.00623.Google Scholar
33Thirring, W.. A soluble relativistic field theory. Ann. Phys. 3 (1958), 91–112.CrossRefGoogle Scholar
34Tian, S.. Initial-boundary value problems for the general coupled nonlinear Schrd̈inger equation on the interval via the Fokas method. J. Differ. Equ. 262 (2017), 506–558.CrossRefGoogle Scholar
35Villarroel, J.. The DBAR problem and the Thirring model. Stud. Appl. Math. 84 (1991), 207–220.CrossRefGoogle Scholar
36Wadati, M. and Sogo, K.. Gauge transformation in soliton theory. J. Phys. Soc. Japan 52 (1983), 394–338.CrossRefGoogle Scholar
37Xia, B.. The Ablowitz-Ladik system on a finite set of integers. Nonlinearity 31 (2018), 3086–3114.CrossRefGoogle Scholar
38Xia, B. and Fokas, A. S.. Initial-boundary value problems associated with the Ablowitz-Ladik system. Physica D 364 (2018), 27–61.CrossRefGoogle Scholar
39Xu, J. and Fan, E.. The unified transform method for the Sasa-Satsuma equation on the half-line. Proc. R. Soc. A 469 (2013), 20130068.CrossRefGoogle ScholarPubMed
40Xu, J. and Fan, E.. Long-time asymptotics for the Fokas-Lenells equation with decaying initial value problem: without solitons. J. Diff. Eqs. 259 (2015), 1098–1148.CrossRefGoogle Scholar
41Zakharov, V. E. and Shabat, A.. A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem, I and II. Funct. Anal. Appl. 8 (1974), 226–35.CrossRefGoogle Scholar
42Zhou, X.. Inverse scattering transform for systems with rational spectral dependence. J. Diff. Eqs. 115 (1995), 277–303.Google Scholar