Third order spectral problems: reductions and Darboux transformations

Abstract

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 lransformation theorems for the reductions corresponding to the above mentioned equations in two spatial and one temporal dimensions are given.

On a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli Equation

“…By virtue of (16), two kinds of the multi-soliton solutions [i.e., (31) and (17)] and another kind of the analytic solutions [i.e., (37)] have been obtained, both with different nonlinear dispersion relations, as illustrated in Figures 1-7. For investigating the integrability of (2), we have obtained Lax Pairs (38) and BTs (44) for (2), with our integrability results as the multi-soliton solutions. 2.…”

“…Equation (4) works as a model for an incompressible fluid [9]. (c) Equation (4) can be regarded as a generalization to some (2+1)-dimensional cases presented in [9,38,39].…”

Transformations of ordinary differential equations via Darboux transformation technique

“…The DT technique is convenient for keeping the reduction constraints imposed on the Lax pair coefficients [27]. We can obtain the potentials to satisfy this reduction, performing the iterations of BDT [25,26].…”

Soliton solutions and Bäcklund transformations of a (2 + 1)-dimensional nonlinear evolution equation via the Jaulent–Miodek hierarchy

“…To understand the phenomena modeled by the NLEEs, some methods have been developed for people to search for the explicit solutions of the NLEEs, such as the inverse scattering transformation [1], Bell polynomials [10][11][12][13][14], Hirota method [15,16], Bäck-lund transformation (BT) [17,18], Darboux transformation [19], and Lie group method [20][21][22][23].…”