A 3 ¡Á 3 Spectral Problem Of The Darboux Transformation And Exact Solutions, Linked To The Soliton Equations

Soliton theory is an important subject of nonliearscience.There are at least two useful applications.First,it reflects a set of rather stable phenomena.Second,it provides a way to get the explicit solutions for nonlinear partial differential equations.Therefore,it has been thinking highly of by physicists and mathematicans. In this paper ,we are going to study a three potential soliton equation[20]It is well known that there are several systematic approaches to obtain solutions of soliton equations. Darboux transformation(DT) has been proved to be one of the most simple and beautiful method to get explicit solutions of some soliton equations from a trivial seed.In section 1, we introduce Darboux transformation theory and Darboux matrix method. Based on these methods, in the following paper, a Darboux transformations are constructed.In section 2, consider Lax pairsAccording to the zero curature equation U_t — V_x + [U, V] = 0 yields the three potential soliton equation.Using the equationTx + TU = UT,( U and U has the same form, except changing u, v,w into u, v,w ) a Darboux transformations with multi-parameters are derived. With choose the Darboux transformationT =n-lE ' i=0"?23i=0E8 = 0n— 1 t=0n-lE-8=0i=0n-ln-lEt=0An+t=0(a-" (i,j = 1,2,3) is the function of x,t. )In section 3, from a trivial seed u = v = w = 0, we use this Darboux transformation to generate new soliton solutions of the three potential soliton equation, then discuss the first two cases. When n = 1,2, suitably choosing parameters, we plot beautiful exlicit solution figures.