Initial boundary value problem for nonlinear Dirac equation of Gross-Neveu type in 1 + 1 dimensions

Abstract

This paper studies an initial boundary value problem for a class of nonlinear Dirac equations with cubic terms and moving boundary. For the initial data with bounded L 2 norm and the suitable boundary conditions, the global existence and the uniqueness of the strong solution are proved.

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

“…Similarly, at different powers of λ in the (2, 1) entries of the Lax equation (15), we obtain two constraints αγV n (|R n+1 | 2 − |R n | 2 ) − (αβ − γδ)V n + αγ(βR n − δR n+1 ) + V 2 n (αR n − γR n+1 ) = 0, (40) βδV n (|Q n | 2 − |Q n+1 | 2 ) − (αβ − γδ)V n + βδ(αQ n+1 − γQ n ) + V 2 n (βQ n+1 − δQ n ) = 0, (41) and the evolution equation…”

“…The standard Cauchy problem is posed in the time coordinates t ∈ R for the initial data (u 0 , v 0 ) extended in the spatial coordinate x ∈ R. Solutions of the Cauchy problem were studied recently in [7,8,14,15,16,31,40,41]. Integrability of the MTM follows from the existence of the following Lax operators:…”

Inverse Scattering for the Massive Thirring Model

“…Indeed, the local and global existence of solutions to the Cauchy problem for the MTM system (1.1) in the L 2 -based Sobolev spaces H m (R), m ∈ N can be proven with the standard contraction and energy methods, see review of literature in [26]. Low regularity solutions in L 2 (R) were already obtained for the MTM system by Selberg and Tesfahun [31], Candy [5], Huh [13,14,15], and Zhang [35,36]. The well-posedness results can be formulated as follows.…”

Large Time Behavior of Solution to Nonlinear Dirac Equation in 1+1 Dimensions

“…The survey of wellposedness and stability results for nonlinear Dirac equations in one dimension is given in [13]. Recently, the global existence of solutions in L 2 for Thirring model in R 1+1 has been established by Candy in [4], while the well-posedness for solutions with low regularity for Gross-Neveu model in R 1+1 has been obtained in Huh and Moon [11], and in Zhang and Zhao [19].…”

“…For any sequence of classical solutions (u (n) , v (n) ) given as in Definition 1.1, it has been shown in [19] that for any T > 0, there holds the following,…”