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

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.