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(±∞)r(±∞)=λ²₀\lessgtr0 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, iq_t+q_xx−mi(|q|²q)_x=0(m=−1, +1), is solved under the nonvanishing boundary condition, |q|²→mλ₀² 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.