Darboux Transformation and Variable Separation Approach: the Nizhnik-novikov-veselov Equation

Darboux transformation is developed to systematically find variable separation solutions for the Nizhnik-Novikov-Veselov equation. Starting from a seed solution with some arbitrary functions, the once Darboux transformation yields the variable separable solutions which can be obtained from the truncated Painlevé analysis and the twice Darboux transformation leads to some new variable separable solutions which are the generalization of the known results obtained by using a guess ansatz to solve the generalized trilinear equation. PACS.02.30.Jr, 02.30.Ik, 05.45.Yv. To find some exact solutions of integrable systems has focused many mathematicians and physicists’ attention since the soliton theory came into being at the 1960’s. There are many important methods to obtain the special solutions of a given soliton equation. Some of the most important methods are the inverse scattering transformation (IST) approach[1], the bilinear form[2], symmetry reduction[3], Bäcklund transformation and Darboux transformation [5] etc. In comparison with the linear case, it is known that IST is an extension of the Fourier transformation in the nonlinear case. In addition to the Fourier transformation, there is another powerful tool called the variable separation method [6] in the linear case. Recently, some kinds of variable separation approaches had been developed to find new exact solutions of nonlinear models, say, the classical method, the differential Stäckel matrix approach [7], the geometrical method [8], the ansatz-based method [9, 8], functional variable separation approach[10], the derivative dependent functional variable separation approach[11], the formal variable separation approach (nonlinearization of the Lax pairs or symmetry constraints) [12] and the informal variable separation methods [13]–[15]. 1 Especially, for various (2+1)-dimensional nonlinear physics models, a quite universal formula U ≡ 2(a1a2 − a0a3)qypx (a0 + a1p+ a2q + a3pq) , (1) where ai, (i = 0, 1, 2, 3) are arbitrary constants and p = p(x, t) and q = q(y, t) are arbitrary constants of the indicated variables, is found by using the informal variable separation approach[13]–[15]. Starting from the universal formula (1), abundant localized excitations like the dromions, lumps, ring solitons, breathers, instantons, solitoffs, fractal and chaotic patterns are found. Now a very important question is can we find the universal formula from other well known powerful methods like the IST approach, dressing method, Darboux transformation (DT) etc? DT is one of the most powerful methods to construct a broad class of considerable physical interest and important nonlinear evolution equations such as the well-known Korteweg-de Vries (KdV) equation, the Kadomtsev-Petviashvili (KP) equation, the Davey-Stewartson (DS) equation, the sine-Gordon (SG) equation [4] and so on. In this letter, we use the Darboux transformation to study the variable separable solutions for the (2+1) dimensional Nizhnik-Novikov-Veselov system.[16] ut = uxxx + uyyy + 3(vu)x + 3(uw)y, (2) ux = vy, uy = wx. (3) The (2+1)-dimensional NNV equation is an only known isotropic Lax integrable extension of the well-known (1+1)-dimensional KdV equation. Many authors have studied the solutions of the NNV equation. For example, Boiti et al[17] solved the NNV equation via the IST; Tagami and Hu and Li obtained the soliton-like solutions of the NNV equation by means of Bäcklund transformation[18]; Hu also gave out the nonlinear superposition formula of the NNV equation[19]; Some special types of multi-dromion 2 solutions were found by Radha and Lakshmanan[20]; The generalized localized excitations expressed by (1) were given in [13] and [15] and the special binary Darboux transformation was given in [4]. It is known that the NNV equation system (2) and (3) can be represented as a compatibility condition of the linear system