Exact Solutions of the Derivative Nonlinear Schrödinger Equation under the Nonvanishing Conditions

Abstract

The inverse method related to a modified Zakharov-Shabat eigen value problem with nonvanishing potentials q ( x ) and r ( x ), where \(q({\pm}\infty)r({\pm}\infty){=}\lambda^{2}_{0}{\lessgtr}0\) is developed. There exists a certain class of the nonlinear evolution equations which are solvable by this method. As the most primitive case a derivative nonlinear Schrödinger equation, i q t + q x x - m i(| q | 2 q ) x =0( m =-1, +1), is solved under the nonvanishing boundary condition, | q | 2 → m λ 2 0 as x →±∞. There appear paired-solitons because of the nonvanishing condition. One paired-soliton solution is obtained with the closed form. This solution shows an algebraic behaviour under a certain limiting condition.