Abstract
We use a generalized Cole‐Hopf transformation to obtain a condition that allows us to find exact solutions for several forms of the general seventh‐order KdV equation (KdV7). A remarkable fact is that this condition is satisfied by three well‐known particular cases of the KdV7. We also show some solutions in these cases. In the particular case of the seventh‐order Kaup‐Kupershmidt KdV equation we obtain other solutions by some ansatzes different from the Cole‐Hopf transformation.
Highlights
During the last years scientists have seen a great interest in the investigation of nonlinear processes. The reason for this is that they appear in various branches of natural sciences and in almost all branches of physics: fluid dynamics, plasma physics, field theory, nonlinear optics, and condensed matter physics
It is clear that the knowledge of closed-form solutions of NLPDEs facilitates the testing of numerical solvers, helps physicists to better understand the mechanism that governs the physic models, provides knowledge of the physic problem, provides possible applications, and aids mathematicians in the stability analysis of solutions
The simplest classes of exact solutions to a given partial differential equation are those obtained from a traveling-wave transformation
Summary
During the last years scientists have seen a great interest in the investigation of nonlinear processes The reason for this is that they appear in various branches of natural sciences and in almost all branches of physics: fluid dynamics, plasma physics, field theory, nonlinear optics, and condensed matter physics. In this sense, the study of nonlinear partial differential equations NLPDEs and their solutions has great relevance today. The simplest classes of exact solutions to a given partial differential equation are those obtained from a traveling-wave transformation.
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