On the long-time asymptotic for a coupled generalized nonlinear Schrödinger equations with weighted Sobolev initial data

Abstract

From the matrix ∂̄ problem, we propose a novel integrable coupled hierarchy with the application of recursive operator Λn which includes a coupled generalized nonlinear Schrödinger (cgNLS) equations with its Lax pair as n=0. We successfully derive infinite conservation laws and Hamiltonian function of the cgNLS equations for the first time. Furthermore, we employ the ∂̄ steepest descent method in order to study the Cauchy problem of the cgNLS equations with initial conditions in weighted Sobolev space H1,1(R)=H1(R)∩L2,1(R). In a fixed space–time cone S(x1,x2,v1,v2)={(x,t)∈R2:x=x0+vt,x0∈[x1,x2],v∈[v1,v2]}, the long-time asymptotic behavior of the solutions u(x,t) and v(x,t) is derived. Based on the results of asymptotic behavior, we prove the soliton resolution conjecture of the cgNLS equations which consist of three terms, the soliton term confirmed by |Z(I)|-soliton on discrete spectrum, the t−12 order term on continuous spectrum and the residual error up to O(t−34).