L²orbital stability of Dirac solitons in the massive Thirring model

Abstract

We prove L 2 orbital stability of Dirac solitons in the massive Thirring model. Our analysis uses local well posedness of the massive Thirring model in L 2 , conservation of the charge functional, and the auto-Bäcklund transformation. The latter transformation exists because the massive Thirring model is integrable via the inverse scattering transform method.

Darboux transformation and soliton solutions of the semi-discrete massive Thirring model

“…Let us check that the discrete solitons (51) recover solitons of the continuous MTM system (2). In order to simplify the computations, we set δ 1 = 1, which corresponds to the case of stationary solitons [6,22]. By defining x n = hn, n ∈ Z and taking the limit h → 0, we obtain for δ 1 = 1:…”

“…The massive Thirring model (MTM) in laboratory coordinates is an example of the nonlinear Dirac equation arising in two-dimensional quantum field theory [23], optical Bragg gratings [7], and diatomic chains with periodic couplings [1]. This model received much of attention because of its integrability [17] which was used to study the inverse scattering [13-16, 21, 27, 28], soliton solutions [2][3][4]20], spectral and orbital stability of solitons [6,12,22], and construction of rogue waves [8].…”

Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation

“…The time evolutions of the charge density from 푡 = 0 to 푡 = 30 are shown in Figure 1. The collision phenomenon of two solitary waves is depicted in Figure 1, from which we can see that the solitary waves keep moving with their original velocities and shapes after the collision [34][35][36].…”

Integrable semi-discretization of the massive Thirring system in laboratory coordinates

“…More recently, the same Lax operators have been used for many purposes, e.g. for the inverse scattering transform [20,21,28,38,39], for spectral stability of solitary waves [18], for orbital stability of Dirac solitons [10,29], and for construction of rogue waves [13]. Numerical simulations of nonlinear PDEs rely on spatial semi-discretizations obtained either with finite-difference or spectral methods.…”