Numerical calculation of N-periodic wave solutions of the negative-order Korteweg-de Vries equations

In this paper, the N-periodic wave solutions of the negative-order Korteweg-de Vries equations are presented, which can be used to describe wave phenomena in the water waves and plasma waves. Combining the bilinear Bäklund transformation with the Riemann-theta function, the N-periodic wave solutions can be obtained. Employing the parity of the bilinear forms for the Bäklund transformation, the complexity of the calculation can be reduced. The difficulty of solving N-periodic wave solutions can be transformed into solving least square problems. The Gauss-Newton numerical algorithm is employed to solve this kind of problem. Furthermore, the characteristic lines are used to analyze quantitatively the quasi-periodic solutions. The characteristic line analysis method is specifically demonstrated in the case of N=3. Some examples of numerical simulations for the 3-periodic and 4-periodic waves are presented. It is proved that this method can be further extended to the N-periodic wave solutions.