Inverse Scattering Method for the Nonlinear Evolution Equations under Nonvanishing Conditions

An extended inverse scattering method is developed to solve the nonlinear evolution equations which are based on the AKNS eigenvalue problem with nonvanishing potentials q ( x ) and r ( x ) where \(q(x)r(x){\rightarrow}\lambda_{0}^{2}({\gtrless}0)\) as x →±∞. As an example, we solved the case of nonlinear Schrödinger equation, i q _t + q _{x x} -2( m | q | ² -λ ₀ ² ) q =0 ( m =-1, +1), under the nonvanishing boundary conditions, q ( x , t )→ q ^± as x →±∞, where q ^± are constants. For m =1 we get the “envelope dark soliton,” while for m =-1 there appears a new solution as the extended form of the “envelope bright soliton.”