Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations

The Moutard transformation is a Darboux-type transformation appropriate to linear scattering problems of hyperbolic or elliptic type in two independent and one dependent variable. The authors derive this transformation, as well as the Darboux transformation for parabolic linear problems, by a factorization method and discuss the use of compositions of such transformations in the construction of solutions to the Novikov-Veselov equations.

http://iopscience.iop.org/0266-5611/7/6/005/pdf/0266-5611_7_6_005.pdf

On the Moutard transformation for integrable partial differential equations

Lie symmetry analysis, exact solutions and conservation laws for the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

Spectral problems of the third order (matrix and scalar) and associated systems of non-linear equations are studied. The technique to construct the solutions is developed for possible reduction constraints. The Dodd-Bullough-Mikhailov, Hirota-Satsuma, Boussinesq, Sawada-Kotera, Kaup-Kupershmidt and less popular reductions of nonlinear systems are treated by means of the Darboux transformation method with the covariant bilinear conditions for the given reduction. Properties of some new soliton-like solutions are discussed. The generalization of Darboux transformation theorems for the reductions corresponding to the above mentioned equations in two spatial and one temporal dimensions are given.

Third order spectral problems: reductions and Darboux transformations

An extended simplest equation method and its application to several forms of the fifth-order KdV equation

The solution of the third-order isospectral equation of the Caudrey-Dodd-Gibbon-Sawada-Kotera equation (CDGSKE) for soliton potential is obtained recursively from the Riccati equation derived by iterating once the auto-Backlund transformation. It is then shown that the discrete eigenfunctions of the sixth-order recursion operator for this equation can be written in terms of the solutions of the isospectral equation. The behaviour of the 1-soliton solution which has certain novel features is studied. A sine-Gordon-like equation resembling the double-sine-Gordon equation is derived from the CDGSKE.

https://iopscience.iop.org/article/10.1088/0305-4470/19/18/022/pdf

W Oevel

Papers

Solitons and discrete eigenfunctions of the recursion operator of non-linear evolution equations. I: The Caudrey-Dodd-Gibbon-Sawada-Kotera equation