Some elliptic traveling wave solutions to the Novikov-Veselov equation

Abstract

An approach is proposed to obtain some exact explicit solutions in terms of elliptic functions to the Novikov-Veselov equation (NVE[V (x, y, t)] = 0). An expansion ansatz V -> psi = Sigma(2)(j=0) a(j)f(j) is used to reduce the NVE to the ordinary differential equation (f') 2 = R(f), where R(f) is a fourth degree polynomial in f. The well-known solutions of (f') 2 = R(f) lead to periodic and solitary wave like solutions V. Subject to certain conditions containing the parameters of the NVE and of the ansatz V.. the periodic solutions V can be used as start solutions to apply the (linear) superposition principle proposed by Khare and Sukhatme.