The Painleve Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients

The general KdV equation (gKdV) derived by T. Chou is one of the famous (1+1) dimensional soliton equations with variable coefficients. It is well-known that the gKdV equation is integrable. In this paper a higher-dimensional gKdV equation, which is integrable in the sense of the Painleve test, is presented. A transformation that links this equation to the canonical form of the Calogero-Bogoyavlenskii-Schiff equation is found. Furthermore, the form and similar transformation for the higher-dimensional modified gKdV equation are also obtained.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Conservation Laws and Travelling Wave Solutions for a Negative-Order KdV-CBS Equation in 3+1 Dimensions

In this paper, we study a new negative-order KdV-CBS equation in (3 + 1) dimensions which is a combination of the Korteweg-de Vries (KdV) equation and Calogero-Bogoyavlenskii-Schiff (CBS) equation. Firstly, we determine the Lie point symmetries of the equation and conservation laws by using the multiplier method. The conservation laws will be used to obtain a triple reduction to a second order ordinary differential equation (ODE), which lead to line travelling waves and soliton solutions. Such solitons are obtained via the modified form of simple equation method and are displayed through three-dimensional plots at specific parameter values to lend physical meaning to nonlinear phenomena. It illustrates that these solutions might be extremely beneficial in understanding physical phenomena in a variety of applied mathematics areas

The Painlevé Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients

The general KdV equation (gKdV) derived by T. Chou is one of the famous (1 + 1) dimensional soliton equations with variable coefficients. It is well-known that the gKdV equation is integrable. In this paper a higher-dimensional gKdV equation, which is integrable in the sense of the Painlevé test, is presented. A transformation that links this equation to the canonical form of the Calogero-Bogoyavlenskii-Schiff equation is found. Furthermore, the form and similar transformation for the higher-dimensional modified gKdV equation are also obtained

Towards Noncommutative Integrable Systems

We present a powerful method to generate various equations which possess the Lax representations on noncommutative (1+1) and (1+2)-dimensional spaces. The generated equations contain noncommutative integrable equations obtained by using the bicomplex method and by reductions of the noncommutative (anti-)self-dual Yang-Mills equation. This suggests that the noncommutative Lax equations would be integrable and be derived from reductions of the noncommutative (anti-)self-dual Yang-Mills equation, which implies the noncommutative version of Richard Ward conjecture. The integrability and the relation to string theories are also discussed.Comment: 13 pages, 1 figure, LaTeX; v2: typos corrected, comments and references added; v3: comments and references added, version to appear in Phys. lett.

Extractions of some new travelling wave solutions to the conformable Date-Jimbo-Kashiwara-Miwa equation

In this paper, complex and combined dark-bright characteristic properties of nonlinear Date-Jimbo-Kashiwara-Miwa equation with conformable are extracted by using two powerful analytical approaches. Many graphical representations such as 2D, 3D and contour are also reported. Finally, general conclusions of about the novel findings are introduced at the end of this manuscript

Some new exact solutions of (4+1)-dimensional Davey–Stewartson-Kadomtsev–Petviashvili equation

Exact solutions of nonlinear equations have got formidable attraction of researchers because these solutions demonstrate the physical behaviour of a model. In this paper, we focus on extracting some new exact solutions of a (4+1)-dimensional Davey–Stewartson-Kadomtsev–Petviashvili (DSKP) equation. To find new travelling wave solutions of the DSKP equation, we use ()-expansion technique. The obtained solutions are in the form of the exponential and trigonometric functions. We obtain different kinds of waves solutions for specific values of parameters. We simulate the achieved solutions in 3D and 2D plots.The authors are grateful to the Basque Government, Spain for its support through Grant IT1555-22 and to MCIN/AEI 269.10.13039/5011 00011033 for Grant PID2021-1235430B-C21/C22. All authors approved the version of the manuscript to be published

Reducing the PDEs to ODEs through lie vectors using the integrated factors

We reduce the PDEs to ODEs through Lie vectors as previously done through two reduction stages. Some of these ODEs have no solution. Some researchers in this step, use the SMM, power series method or Riccati equation method to solve non-solvable equations. We use the integrating factors as a tool to reduce the order and the nonlinearity in an ODE. This explores new solutions as it appears for the (2+1)-dimensional (CBS) and (3+1)-dimensional generalized BKP solutions compared results