The (2+1)‐dimensional Schwarzian Korteweg–de Vries equation and its generalizations with discrete Lax matrices

Abstract

By using the discrete Lax matrices corresponding to and in the Adler–Bobenko–Suris list of quadrilateral lattice equations, we establish solutions of the (2+1)‐dimensional Schwarzian Korteweg–de Vries (SKdV) equation and its generalizations. According to the structure of integrable Hamiltonian systems provided by one discrete Lax matrix for the lattice, we construct a novel Lax representation for the (2+1)‐dimensional SKdV equation. On the basis of the Riemann surface and elliptic variables, the Hamiltonian phase flows related to the equation are linearized by the Abel–Jacobi coordinates. Consequently, we obtain finite genus solutions to the (2+1)‐dimensional SKdV equation. From the discrete Lax matrix of the lattice, we study the generalized (2+1)‐dimensional SKdV equation with a parameter , which can be reduced to a simple case of the Krichever–Novikov equation. Finally, we show that the case of can also be solved through a modified version of the other discrete Lax matrix for the equation.