Exact Traveling Wave Solutions For Four Nonlinear Integral Differential Equations

Nonlinear partial differential equations play an important role in both theory and practice.The study of traveling wave solution can help people to understand the change law of motion and the natural phenomenon better.The purpose of this disserta-tion is to discuss the exact traveling wave solutions of four nonlinear partial differential equations by using the auxiliary equation method,the e-?(?)-expansion method,the extended test equation method and the Riccati equation expansion method,the specific four nonlinear partial differential equations have the following forms:1.The(1+1)-dimensional integro-differential Ito equation:utt+uxxxt+3(2uxut+uuxt)+3uxx(?)x-1(ut)=0,(?)2.The(2+1)-dimensional integro-differential Sawada-Kotera equation:ut=(uxxxx+5uuxx+5/3u3+uxy)x-5(?)x-1(uyy)+5uuy+5ux(?)x-1(uy),(?)3.The first integro-differential Kadomtsev-Petviashvili hierarchy equation:ut=1/2uxxy+1/1(?)x-2(uyyy)+2ux(?)x-1(uy)+4uuy,(?)4.The second integro-differential Kadomtsev-Petviashvili hierarchy equation:ut=1/16uxxxxx+5/4(?)x-1(uuyy)+5/4(?)x-1(uy2)+5/16(?)x-3(uyyyy)+5/4ux(?)x-2(uyy)(?)+5/2u(?)-1x(uyy)+5/2uy(?)x-1(uy)+15/2u2ux+5/2uxuxx+5/4uuxxx-5/8uxyy.By studying the traveling wave solutions of the above equations,the following conclusions are obtained:1.By applying integral substitution and traveling wave transformation,an ordinary differential equation with fifth order in corresponding to equation(?)is obtained.After using variable substitution and integral operation,the fifth order ordinary differential equation can be turned into the second order ordinary differential equation.We carry out the theory and method of planar dynamical system for the equivalent dynamical system,the conclusions of qualitative analysis for traveling wave solutions of equation(?)are studied,there exist two bell-shaped solitary wave solutions and many bounded traveling wave solutions.With the help of auxiliary equation method under different conditions,the exact expressions of bell-shaped solitary wave solutions and some bounded traveling wave solutions of equation(?)are investigated2.By applying integral substitution and traveling wave transformation,an ordinary differential equation with sixth order in corresponding to equation(?)is obtained.After using variable substitution and integral operation,the sixth order ordinary differential equation can be turned into the fourth order ordinary differential equation.With the help of the e-?(?)-expansion method,the problem of solving ordinary differential equations is transformed into the problem of solving algebraic equations.According to the solutions of ?(?)in different situations,hyperbolic solution,trigonometric solution and bounded traveling wave solution of equation(?)are obtained.Finally,we give the shape of these solutions by Maple software3.By applying integral substitution and traveling wave transformation,an ordinary differential equation with fourth order in corresponding to equation(?)is obtained.After using variable substitution and integral operation,the fourth order ordinary differential equation can be turned into the second order ordinary differential equation.We carry out the theory and method of planar dynamical system for the equivalent dynamical system,the conclusions of qualitative analysis for traveling wave solutions of equation(?)are studied,there exist the bell-shaped solitary wave solutions and infinite periodic wave solutions.With the help of the extended test equation method,We obtain the explicit expressions of the bell-shaped solution,trigonometric solution,hyperbolic solution and elliptic solution of equation(?)4.By applying integral substitution and traveling wave transformation,an ordinary differential equation with eighth order in corresponding to equation(?)is obtained.After using variable substitution and integral operation,the eighth order ordinary differential equation can be turned into the fourth order ordinary differential equation.By using the Riccati equation expansion method,the problem of solving ordinary differential equations is transformed into the problem of solving algebraic equations.According to solutions of Riccati equation in different cases,the exact expressions of kink-shaped,bell-shaped solutions,trigonometric function solutions and hyperbolic function solution to equation(?)are investigated.With the aid of Maple software,the shape and behavior of these solutions are given.