The Exact Solution Of The Kp And The Coupling Of The Generalized H-s Kdv Equations

The soliton theory is an important constituent part of applied mathematics and mathematical physics, and an important branch of the nonlinear science. There are many methods to obtain exact solutions of nonlinear partial differential equations in the soliton theory. Among the various approaches, the Darboux transformation and the bilinear method are two very powerful tools, which can find exact solutions of soliton equations .For this purpose,we make an endeavor to studying and discussing the following two aspects:First, applying to the theory of the Darboux transformation, we consider the 2×2 a Lax pairsΦ_x = UΦ,Φ_y = V₁ΦΦ_t = V₂φand the corresponding (2+1)-dimension KP soliton equation :w_t = 3/4(?)_x^-1w_yy + (1/16w_xx - 3/2w²)_x.We construct the Darboux transformation of them and give a complete proof of it.We get mult-soliton of the nonlinear equations especially obtain the exact solutions of the KP equation through the transformation of w(x,y,t) = (v(x,y,t))³ + r₁. Through choosing u = 0, v≠0 as a seed solution, we discuss the first two simple cases N=1,N=2. Suitably choosing parameters,we plot the exact soliton figures.Second, through Bi-logarithmic transformation u = - (ln f)_xx,v = g/f,w = h/f, generalization of the Hirota-Satsuma Coupled KdV soliton equation can be transformation into bilinear differential equationsFurther, we find the exact soliton solutions by a perturbation method and then plot soliton figures.