On quasiperiodic wave solutions and integrability to a generalized $$\varvec{(2+1)}$$ ( 2 + 1 ) -dimensional Korteweg–de Vries equation

Abstract

Under investigation in this paper is a generalized \((2+1)\)-dimensional Korteweg–de Vries equation, which could describe many nonlinear phenomena in plasma physics. By virtue of the Bell’s polynomials, a straightforward way is presented to succinctly construct its bilinear form, bilinear Backlund transformation and Lax pairs. Once the Lax pairs obtained, the important infinite conservation laws of the equation are directly found. Moreover, based on the bilinear formalism, we construct the Riemann theta function periodic wave solutions and soliton solutions. Finally, the relationships between the periodic wave solutions and soliton solutions are strictly established, and the asymptotic behavior of the periodic waves is also presented with detailed proof.