Darboux transformations for the nonlinear Schrödinger equations

Mathematics, 2021

In this paper, by using the gauge transformation and the Lax pairs, the N-fold Darboux transformation (DT) of the classical three-component nonlinear Schrödinger (NLS) equations is given. In addition, by taking seed solutions and using the DT, exact solutions for the given NLS equations are constructed.

Journal of Physics A: Mathematical and General, 2004

We derive generalized nonlinear wave solution formula for mixed coupled nonlinear Schödinger equations(mCNLSE) by performing the unified Darboux transformation. We give the classification of the general soliton formula on the nonzero background based on the dynamical behavior. Especially, the conditions for breather, dark soliton and rogue wave solution for mCNLSE are given in detail. Moreover, we analysis the interaction between dark-dark soliton solution and breather solution. These results would be helpful for nonlinear localized wave excitations and applications in vector nonlinear systems. 1 Introduction Nonlinear Schrödinger equation (NLSE) is an important model in mathematical physics, which can be applied to hydrodynamics [1], plasma physics [2], molecular biology [3] and optics [4]. Recently, Peregrine soliton (rogue wave solution), Akhmediev breather, Kuznetzov-Ma breather and dark soliton were observed in experiments in succession. For instance, Kuznetzov-Ma soliton was confirmed in 2012 [3], the Akhmediev breather was verified in numerical experiment [5], the Peregrine soliton was experimentally observed in nonlinear fibre optics system [6], water tank [7, 8] and plasma [9]. Dark soliton was observed on the surface of water [10]. Indeed those exact solutions for the NLSE on the plane wave background were known well long time ago [3]. For the focusing NLSE, there exists Akhmediev breather, Kuznetzov-Ma soliton and Peregrine soliton. There exists dark soliton for the defocusing NLSE.

Il Nuovo Cimento B, 1981

2022

In this article, we construct the Darboux solutions of integrable coupled nonlinear wave equation associated with Hirota Satsuma system in Darboux framework with their N-th generalization in terms of Wronskians through its Lax pair. We also derive the exact solitonic solutions for the coupled eld variables of that system with the help of one and twofold Darboux transformations in the background of zero seed solution. This work also encloses the derivation of zero curvature representation for the integrable coupled nonlinear waves equation possessing traceless matrices through its existed Lax pair, which may be assumed to t in AKNS scheme as it usually involves the order 2 traceless matrices.

Journal of Nonlinear Mathematical Physics, 2021

We consider Darboux transformations for the derivative nonlinear Schrödinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant forms. Additionally, the parabolic and soliton solutions of the derivative nonlinear Schrödinger equation are given as explicit examples.

Chaos, Solitons & Fractals, 2009

In this paper, we propose a new approach (different from the approach presented in Proc. R. Soc. Lond. A 460 2617-2627) to calculate multi-soliton solutions of Camassa-Holm equation (CH) and modified Camassa-Holm (MCH) equation with aid of Darboux transformation (DT). We first map the CH and MCH equation to a negative order KdV (NKdV) equation by a reciprocal transformation. Then we proceed to apply the DT to solve the NKdV equation in the usual way. Finally we invert the reciprocal transformation to recover the solutions of the CH equation and MCH equation.

Topics in Operator Theory, 2010

A certain symmetry is exploited in expressing exact solutions to the focusing nonlinear Schrödinger equation in terms of a triplet of constant matrices. Consequently, for any number of bound states with any number of multiplicities the corresponding soliton solutions are explicitly written in a compact form in terms of a matrix triplet. Conversely, from such a soliton solution the corresponding transmission coefficients, bound-state poles, bound-state norming constants and Jost solutions for the associated Zakharov-Shabat system are evaluated explicitly. These results also hold for the matrix nonlinear Schrödinger equation of any matrix size.

Journal of Physics A: …, 2010

A certain symmetry is exploited in expressing exact solutions to the focusing nonlinear Schrödinger equation in terms of a triplet of constant matrices. Consequently, for any number of bound states with any number of multiplicities the corresponding soliton solutions are explicitly written in a compact form in terms of a matrix triplet. Conversely, from such a soliton solution the corresponding transmission coefficients, bound-state poles, bound-state norming constants and Jost solutions for the associated Zakharov-Shabat system are evaluated explicitly. These results also hold for the matrix nonlinear Schrödinger equation of any matrix size.

Physica D: Nonlinear Phenomena, 1998

We investigate a class of N coupled discretized nonlinear Schriidinger equations of interacting chains in ;I nonlinear lattice. which, in the limit of zero coupling. become integrable Ablowitz-Ladik differential-difference equation\. We \tudl the existence of stationary localized excitations, in the form of soliton-like time-periodic states. by reducing the system to a perturbed 'N-dimensional symplectic map, whose homoclinic orbits are obtained by a recently developed Mcl'nikov analysis. We find that, depending on the perturbation, homoclinic orbits can be accurately located from the simple /eroh oi'a Mel'niko\ vector and illustrate our results in the cases N = 2 and 3.

The European Physical Journal B, 2009

In this paper, analytically investigated is a higher-order dispersive nonlinear Schrödinger equation. Based on the linear eigenvalue problem associated with this equation, the integrability is identified by admitting an infinite number of conservation laws. By using the Darboux transformation method, the explicit multi-soliton solutions are generated in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.