Integrability and Exact Solutions of the (2+1)-dimensional KdV Equation with Bell Polynomials Approach

In this paper, the bilinear formalism, bilinear Bäcklund transformations and Lax pair of the (2+1)-dimensional KdV equation are constructed by the Bell polynomials approach. The N-soliton solution is derived directly from the bilinear form. Especially, based on the two-soliton solution, the lump solution is given out analytically by taking special parameters and using Taylor expansion formula. With the help of the multidimensional Riemann theta function, multiperiodic (quasiperiodic) wave solutions for the (2+1)-dimensional KdV equation are obtained by employing the Hirota bilinear method. Moreover, the asymptotic properties of the one- and two-periodic wave solution, which reveal the relations with the single and two-soliton solution, are presented in detail.