Asymptotics behavior for the integrable nonlinear Schrödinger equation with quartic terms: Cauchy problem

Abstract

We consider the Cauchy problem of integrable nonlinear Schrödinger equation with quartic terms on the line. The first part of the paper considers the Riemann-Hilbert formula via the unified method(also known as the Fokas method). The second part of the paper establishes asymptotic formulas for the solution of initial value problem using the nonlinear steepest descent method(also known as the Deift-Zhou method).

Keywords:

• Integrable nonlinear Schrödinger equation with quartic terms
• Long-time asymptotics
• Nonlinear steepest descent method

2000 Mathematics Subject Classification:

• 37K15
• 41A60
• 35P25