Painlevé analysis and new analytical solutions for compound KdV–Burgers equation with variable coefficients

Abstract

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.