Integrability, lump solutions, breather solutions and hybird solutions for the (2+1)-dimensional variable coefficient Korteweg-de Vries equation

Abstract This paper focuses on the integrability and exact solutions of a (2+1)-dimensional variable coefficient Korteweg-de Vries equation. The bilinear form, Bäcklund transformations, and Lax pair of this equation are obtained using the Bell polynomial method. Soliton solutions, including lump solitons, breather solitons, and hybrid solutions, are constructed by assuming different auxiliary functions in the bilinear ansatz method. Additionally, the soliton solutions are presented as figures for different variable coefficient functions and undetermined items under appropriate parameter choices. The Bäcklund transformations also lead to Lax pair and the infinity conservation laws that ensure the integrability of the nonlinear system under study.