Painlevé analysis, Lax pair, Bäcklund transformation and multi-soliton solutions for a generalized variable-coefficient KdV–mKdV equation in fluids and plasmas

In this paper, a variable-coefficient non-isospectral Korteweg–de Vries–modified Korteweg–de Vries equation arising in fluids and plasmas is investigated. The integrability of such an equation is studied with Painlevé analysis. Under the integrable condition obtained, the Lax pair is also established through the Ablowitz–Kaup–Newell–Segur procedure. The equation is transformed into its bilinear form by virtue of which the multi-soliton/breather solutions and Bäcklund transformation are derived. Soliton propagation, multi-soliton, soliton–breather and breather–breather interactions are studied: different types of solitary waves can be seen with the change of variable coefficients, the existence of compression or broadening depends on the sign of the non-uniformity coefficient, and during the soliton–breather interaction, the propagating direction of the breather is not influenced by the elevation (positive amplitude) or depression (negative amplitude) soliton.