Study Of Explicit Solutions Of Nonlinear Evolution Equations

In this paper, by using the inverse scattering method, the Lie group method, the Darboux transformation and function mappings, we obtain some explicit solutions of nonlinear evolution equations and discuss properties of the solutions.In Chapter two, we discuss the explicit solutions of nonlinear Schrodinger type equations. For the higher-order nonlinear Schrodinger equation (HNSE), our main results are: the heteroclitic, homoclinic and closed curves on the Poincare phase plane correspond to the dark, bright solitary and periodic solutions, respectively; the explicit expressions of these solutions are given. By analysing the effect of higher-order nonlinear and three-order dispersion terms of HNSE,we obtain the following properties: (1) The peak intensity of the solutions is proportional to the ratio of the third-order dispersion and higher- order nonlinear termsinstead of the ratio of the second-order dispersion and nonlinear terms; (2) Whether a bright or dark soliton exists in a monomode optical fiber is determined by the sign of the third-order dispersion k'" instead of the sign of k' , bright soli-tons can exist in the negative third-order dispersion region and dark solitons in the positive third-order dispersion region in a monomode optical fiber, and both bright and dark solitons can propagate not only in the anomalous region, but also in the normal region; (3) Due to the third-order dispersion, the velocity of the soli-ton is modified by for the dark soliton and for the brightsoliton,respectively; (4) Because the velocity of the soliton depends on the soliton width, there do not exist bound N-soliton states. Applying the inverse scattering method and the Darboux transformation, we give the formula of finding multisoli-ton solutions and obtain the 1- soliton,2-soliton,dark soliton and explicit solutions of the Gross-Pitaevskii equation. To find solutions of the N-component nonlinear Schrodinger and Klein-Gordon equations , we generalize the tanh-method to the sec1-tanh1-method, and give explicit solitary wave solutions of the equations in the following forms: , where u = secg( ) , v = tanh1( ).In Chapter three, our main results are the following: (1) We give new explicit and soliton solutions of the (2+1)-dimensional Broer-Kaup equations by applying the homogeneous blance method, (2) Applying the Lie group method, we give some similarity solutions of the (2+1)-dimensional Broer-Kaup equations; (3) Forthe 1 + 1-dimensional and 2+1-dimensional higher-order Broer-Kaup equations , we obtain solitary wave solutions, infinite rational function solutions and closed form solutions.In Chapter four, by using the Lie group method, some solitary wave solutions to a general variable-coefficient KdV equation with dissipative loss and nonuni-formity terms are obtained . Since there exists the damp term, the solitary wave solutions have the property of dissipative attenuation. A special variable-coefficient KdV equation is transformed to a Painleve II equation, which has a solution expressed by the Painleve transcendental function. Applying the generalized tanh-method, we obtain the solitary wave and periodic solutions of some fifth-order nonlinear equations (KdV,SK,KK and GDGSK equations). In addition, we also get some solitary wave solutions and periodic solutions represented by elliptic functions of the coupled KdV systems. The solitary wave solutions have the following properties: the width is constant while the velocity and the pulse amplitude change; Some solutions which are the sum of two classes of solitary waves have much obvious local character.In Chapter five, we show that there exist globally smooth solutions and blowup solutions of the n-dimensional Landau-Lifshitz equations with a external magnetic field. First, we give two types of explicit solutions of the equations. Secondly, we discuss the following properties of the solutions: (1) The curvature k\ of the spatial curve for a class of solutions is k1 = [(k21 + k22)(c')2 + k22r2(1-n)c-2(T - A1t-2]1/2, and it can be seen that k...