【作者】 套格图桑；

【导师】 斯仁道尔吉；

【作者基本信息】 内蒙古师范大学 ， 应用数学， 2004， 硕士

【摘要】 两类辅助方程及其非线性发展方程(组)的精确孤立波解 本文在齐次平衡法，双曲正切函数法和辅助方程法的基础上引入两类辅助方程并利用符号计算系统Mathematica或Maple构造了非线性发展方程(组)的新精确孤立波解。在第一章中我们利用第一类辅助方程(Ⅰ-1)，(Ⅱ-1)，(Ⅲ-1)，(Ⅳ-1)以及把常微分方程的解取为(5)—(6)等两种形式构造了只含有奇阶、偶阶或混合阶数的部分非线性发展方程(组)的新的精确孤立波解。 第一节中构造了Davey-Stewartson(DS)方程组，Kuperschmidt方程，非线性长波方程组，刘维尔方程等非线性发展方程(组)的新类型精确孤立波解。第二节中构造了广义Zakharav-Kaznetsov方程，Boussinesq方程，Modified Kadomtsev-Petviashvili(mKP)方程组等非线性发展方程(组)的精确孤立波解。也构造了齐次平衡法没有直接有效的sine-Gordon型方程，非线性耦合Schrdinger-KdV方程组等非线性发展方程(组)的精确孤立波解。第三节中把解取为(5)—(6)等两种形式构造了(2+1)维Benjamin-Bona-Mahoney(BBM)方程，二维对称正则长波方程组，Pochhammer-Chree方程等非线性发展方程(组)的精确孤立波解。第四节中构造了Benjamin方程，(3+1)维K-P方程，广义对称正则长波方程组，Modified-Benjamin-Bona-Mahoney(mBBM)方程等非线性发展方程(组)的新类型精确孤立波解。 第二章中利用一种双曲函数型假设(或三角函数型假设)和第二 摘要类辅助方程(V一l)，(VI一l)，(VII一1)，(IX一1)构造了二维色散长波方程组，(2+l)维Korteweg一de Vries(Kdy)方程组，Benjamin一Bona-Mahoney(BBM)方程，Modified一Benjamin一Bona一Mahoney(mBBM)方程，Modified一Korteweg一de v.rses(mKdV)方程，Modlifled Kadomtsev一Petviashvili(mKP)方程组，非线性薛定愕(NLS)方程，sine一Gordon型方程等非线性发展方程(组)的新精确孤立波解。 第三章中利用一种指数函数型变换构造了组合KdV-mKdV方程，Fisher方程，非线性电波方程，非线性Klein一Gordon方程等非线性发展方程的新精确孤立波解。更多还原

【Abstract】 In this paper, based on the homogeneous balance method, the tanh-function method and the auxiliary equation method, we have constructed some new exact solitary wave solutions to the nonlinear evolution equations by using two types of auxiliary equations and with the help of symbolic computation system, such as Mathematica or Maple. In chapter 1, using first type of auxiliary equation (I -1),(II -1),(III-1),(IV -1) and taking (5)-(6)as the solutions of the ordinarg differential equation, we constructed the new exact solitary wave solutions to some nonlinear evolution equations which contain odd-order, even-order or the mixed orderderivative terms. In the first section wecontrncted the new exact soliturg wave solutions of the Davey-Stewartson (DS) equations, the Kuperschmidt equation, nonlinear long wave equation, Liouville equation etc.In the second section, we constructed the new exact solitary wave solutions of some nonlinear evolution equations for example, the generalized Zakharov-Kutnetsov equation, Boussinesq equation, Modified Kadomstev-Petviashvili (mKP) equations etc. We also find some exact solitary wave solutions for sme-Gordorn type equations, nonlinear coupled Schrodinger-KdV equations that cannot be solved by using the homogeneous balance method. In the third section, the exactsolitary wave solutions of the 2+1 dimensional Benjamin-Bona-Mahoney (BBM) equation,2+l dimensional symmetric regular long wave equation and the Pochhammar-Chree equation are presented. In the fourth section ,we constructed the new types of exact solitary wave solutions of the Benjamin equation, 3+1 dimensional KP equation, generalized symmetric regularized long wave equations, Modified-Benjamin-Bona -Mahoney (mBBM) equations etc. In chapter 2, using a kind of hyperbolic-function assumption or trigonometric function assumption and the second type of auxiliary equations (V-1),(VI-1),(VI]-1),(IX-1) the new exact solitary wave solutions of the 2+1 dimensional dispersive long wave equations, 2+1 dimensional Korteweg-de Vries(KdV) equation and Benjamin-Bona-Mahoney(BBM) equation Modified-Korteweg-de Vries (mKdV) equation, Modified Kadomstev-Petviashvili(mKP) equation, nonlinear SchrOdinger equation and sine-Gordon type equation arc presented In chapter 3, the new exact solitary wave solutions of the KdV-mKdV equation, Fisher equation, nonlinear telegraphic equation, nonlinear Klein-Gordon equation are obtained by using an exponential type transformation.更多还原