Long-time asymptotic behavior of the modified Schrödinger equation via Dbar-steepest descent method

Abstract

In this paper, we consider the Cauchy problem for the modified NLS equationwhere H 2,2 (R) is a weighted Sobolev space. Using nonlinear steepest descent method and combining the ∂-analysis, we show that inside any fixed conethe long time asymptotic behavior of the solution u(x, t) for the modified NLS equation can be characterized with an N (I)-soliton on discrete spectrum and leading order term O(t −1/2 ) on continuous spectrum up to an residual error order O(t −3/4 ).

Soliton Resolution for the Short-pluse Equation

“…The proof of above proposition is similar to that in [40,42]. In addition, from the definition of k(z) in (4.3) and the symmetry of r(z), we have that k(z 0 ) = k(−z 0 ).…”

“…It has pole at z n and a removable singularity at zn . The calculation of it is similar as it in [42].…”

“…But unlike those of NLS and derivative NLS euqations [38][39][40]42], the Lax pair (2.1) for SP equation has singularities at z = 0 and z = ∞. In order to control the behavior of solutions of (2.1) and construct the solution u(x, t) of the SP equation (1.1), we need use the t-part and the expansion of the eigenfunction as spectral parameter z → 0.…”