Skip to main content Accessibility help

The massive Thirring system in the quarter plane

  • Baoqiang Xia (a1)


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.



Hide All
1Ablowitz, M. J. and Musslimani, Z. H.. Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110 (2013), 064105.
2Ablowitz, M. J. and Musslimani, Z. H.. Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity 29 (2016), 915946.
3Ablowitz, M. J. and Musslimani, Z. H.. Integrable nonlocal nonlinear equations. Stud. Appl. Math. 139 (2017), 759.
4Biondini, G. and Bui, A.. The Ablowitz-Ladik system with linearizable boundary conditions. J. Phys. A: Math. Theor. 48 (2015), 375202.
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.
6Deift, P. and Zhou, X.. A steepest descent method for oscillatory Riemann-Hilbert problems. Ann. Math. 137 (1993), 295368.
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).
8Fokas, A. S.. A unified transform method for solving linear and certain nonlinear PDEs. Proc. R. Soc. London, Ser. A 53 (1997), 1411.
9Fokas, A. S.. Integrable nonlinear evolution equations on the half-line. Commun. Math. Phys. 230 (2002), 139.
10Fokas, A. S.. The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs. Commun. Pure Appl. Math. 58 (2005), 639670.
11Fokas, A. S.. A Unified Approach to Boundary Value Problems,vol. 27 (Philadelphia: Society for Industrial and Applied Mathematics, 2008).
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), 738764.
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.
14Fokas, A. S., Its, A. R. and Sung, L. Y.. The nonlinear Schrödinger equation on the half-line. Nonlinearity 18 (2005), 17711822.
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), 310346.
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).
17Kaup, D. J. and Lakoba, T. I.. The squared eigenfunctions of the massive Thirring model in laboratory coordinates. J. Math. Phys. 37 (1996), 308323.
18Kaup, D. J. and Newell, A. C.. On the Coleman correspondence and the solution of the Massive Thirring model. Lett. Nuovo Cimento 20 (1977), 325331.
19Kawata, T., Morishima, T. and Inoue, H.. Inverse scattering method for the two-dimensional massive Thirring model. J. Phys. Soc. Japan 47 (1979), 13271334.
20Kuznetzov, E. A. and Mikhailov, A. V.. On the complete integrability of the two-dimensional classical Thirring model. Theor. Math. Phys. 30 (1977), 193200.
21Lee, J. H.. Solvability of the derivative nonlinear Schrödinger equation and the massive Thirring model. Theoret. Math. Phys. 99 (1994), 617621.
22Lenells, J.. Initial-boundary value problems for integrable evolution equations with 3 × 3 Lax pairs. Physica D 241 (2012), 857875.
23Lenells, J.. The Degasperis-Procesi equation on the half-line. Nonlinear Anal. 76 (2013), 122139.
24Martinez Alonso, L.. Soliton classical dynamics in the sine-Gordon equation in terms of the massive Thirring model. Phys. Rev. D 30 (1984), 25952601.
25Mikhailov, A. V.. Integrability of the two-dimensional Thirring model. JETP Lett. 23 (1976), 320323.
26Monvel, A. B., Fokas, A. S. and Shepelsky, D.. Integrable nonlinear evolution equations on a finite interval. Commun. Math. Phys. 263 (2006), 133172.
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), 8386.
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), 455458.
29Orfanidis, S. J.. Soliton solutions of the massive Thirring model and the inverse scattering transform. Phys. Rev. D 14 (1976), 472478.
30Pelinovsky, D. E. and Saalmann, A.. Inverse scattering for the massive Thirring model. arXiv: 1801.00039.
31Prikarpatskii, A. K.. Geometrical structure and Bäcklund transformations of nonlinear evolution equations possessing a Lax representation. Theoret. Math. Phys. 46 (1981), 249256.
32Saalmann, A.. Long-time asymptotics for the massive Thirring model. arXiv: 1807.00623.
33Thirring, W.. A soluble relativistic field theory. Ann. Phys. 3 (1958), 91112.
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), 506558.
35Villarroel, J.. The DBAR problem and the Thirring model. Stud. Appl. Math. 84 (1991), 207220.
36Wadati, M. and Sogo, K.. Gauge transformation in soliton theory. J. Phys. Soc. Japan 52 (1983), 394–338.
37Xia, B.. The Ablowitz-Ladik system on a finite set of integers. Nonlinearity 31 (2018), 30863114.
38Xia, B. and Fokas, A. S.. Initial-boundary value problems associated with the Ablowitz-Ladik system. Physica D 364 (2018), 2761.
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.
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), 10981148.
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.
42Zhou, X.. Inverse scattering transform for systems with rational spectral dependence. J. Diff. Eqs. 115 (1995), 277303.


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed