Darboux Transformation and N-Soliton Solution for the Coupled Modified Nonlinear Schrödinger Equations

Abstract

The pulse propagation in the picosecond or femtosecond regime of birefringent optical fibers is governed by the coupled mixed derivative nonlinear Schrödinger (CMDNLS) equations. A new type of Lax pair associated with such coupled equations is derived from the Wadati-Konno-Ichikawa spectral problem. The Darboux transformation method is applied to this integrable model, and the N-times iteration formula of the Darboux transformation is presented in terms of the compact determinant representation. Starting from the zero potential, the bright vector N-soliton solution of CMDNLS equations is expressed as a compact determinant by N complex eigenvalues and N linearly independent eigenfunctions. The collision mechanisms in two components shows that bright vector solitons can exhibit the standard elastic and inelastic collisions. Such energy-exchange collision behaviours have potential applications in the construction of logical gates, the design of fiber directional couplers, and quantum information processors.

Breather and its interaction with rogue wave of the coupled modified nonlinear Schrödinger equation

“…( 1) under the condition m 1 = m 2 . Based on the above initial solutions (4), the one-breather can be given in the following expression by setting ζ = ζ 1 via the traditional Darboux transformation: [23] u…”

“…Equation ( 1) is the complete integrability in view of admitting the N-soliton solution via Dar-boux transformation, and dark and anti-dark vector solitons through the Hirota method reported in former literature. [22][23][24] The double-pole soliton and first-order rogue wave for Eq. ( 1) have also been discussed.…”

Localized Waves for the Coupled Mixed Derivative Nonlinear Schrödinger Equation in a Birefringent Optical Fiber

“…( 1) have been achieved in Refs. [30][31][32][33][34]. Yan obtained the first-order localized wave solutions via DT and Darboux-dressing transformation, including breather wave solution, rogue wave and bright-dark rogue wave [30].…”

A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half‐line

“…The system (2) has been studied extensively on the integrability related to explicit form of the Lax pair and infinitely many local conservation laws, 35 bilinear representations, [35][36][37] soliton solutions given by the Hirota's bilinear method, [35][36][37][38] and rogue waves solutions given by the Darboux transformation method. 39 Recently, the IBV problem of the Manakov system (3) 21,22 and the IBV problem of the coupled derivative NLS equations (4) 27 are also studied by means of the Fokas approach, respectively. To the best of our knowledge, the IBV problem of the system (2) has not been studied.…”