We consider the solutions of the compound Korteweg–de Vries (KdV)–Burgers equation with variable coefficients (vccKdV–B) that describe the propagation of undulant bores in shallow water with certain dissipative effects. The Weiss–Tabor–Carnevale (WTC)–Kruskal algorithm is applied to study the integrability of the vccKdV–B equation. We found that the vccKdV–B equation is not Painlevé integrable unless the variable coefficients satisfy certain constraints. We used the outcome of the truncated Painlevé expansion to construct the Bäcklund transformation, and three families of new analytical solutions for the vccKdV –B equation are obtained. The dispersion relation and its characteristics are illustrated. The stability for the vccKdV–B equation is analyzed by using the phase portrait method.
In this paper, we investigate the solitary wave solutions for the two-dimensional modified Korteweg–de Vries–Burgers (mKdV-B) equation in shallow water model. Despite that Painlevé test fails to prove the integrability of the mKdV-B equation by using the WTC-Kruskal algorithm, the Bäcklund transformation is obtained via the truncation expansion. The exact solutions of the mKdV-B equation are found using factorization techniques, Exp-function and energy integral approach of the corresponding ordinary differential equation. We found that the dispersion relation of the linearized mKdV-B equation lies on the complex plane yielding a damping character. By keeping the water height relatively small, we illustrate the resulting solutions in several figures showing the shock and solitary wave nature in the flow. The stability for the mKdV-B equation is analyzed by using the phase plane method.
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