Solitary waves with the Madelung fluid description: A generalized derivative nonlinear Schrödinger equation

The Darboux transformation (DT) for the integrable nonlocal nonlinear Schrodinger equation (nNLSE) is constructed with the aid of loop group method. Based on the DT, we derive several types of analytical solutions for the focusing case and defocusing case, such as the periodical solutions, soliton solutions, breathers and rational ones. Dynamical properties for those solutions to the nNLSE are analyzed on basis of the solution expressions and figures.

Soliton solutions for the nonlocal nonlinear Schrödinger equation

Analytical study on a two-dimensional Korteweg–de Vries model with bilinear representation, Bäcklund transformation and soliton solutions

Within the framework of the Madelung fluid description, in the present paper, we will derive bright and dark (including gray- and black-soliton) envelope solutions for a generalized mixed nonlinear Schrodinger model
 $${\mathrm {i}}\,\dfrac{\partial \varPsi }{\partial t}=\dfrac{\partial ^2 \varPsi }{\partial x^2}+{\mathrm {i}}\,a\,|\varPsi |^{2}\,\dfrac{\partial \varPsi }{\partial x}+{\mathrm {i}}\,b\,\varPsi ^{2}\,\dfrac{\partial \varPsi ^*}{\partial x}+c\,|\varPsi |^{4}\varPsi +d\,|\varPsi |^{2}\varPsi $$
 , by virtue of the corresponding solitary wave solutions for the generalized stationary Gardner equations. Via corresponding parametric constraints, our results are achieved under suitable assumptions for the current velocity associated with different boundary conditions of the fluid density $$\rho $$
 , while we have only considered the motion with stationary-profile current velocity case and excluded the motion with constant current velocity case. Note that our model is a generalized one with the inclusion of multiple coefficients (
 $$a$$
 , $$b$$
 , $$c$$
 and $$d$$
 ).

Madelung fluid description on a generalized mixed nonlinear Schrödinger equation

We construct an analytical and explicit representation of the Darboux transformation (DT) for the KunduEckhaus (KE) equation. Such solution and n-fold DT Tn are given in terms of determinants whose...

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2015.0236

Mingshu Peng

Papers

Systematic construction of infinitely many conservation laws for certain nonlinear evolution equations in mathematical physics