非线性发展方程的因式分解方法
Factorization method for nonlinear evolution equations Factorization method for nonlinear evolution equations
摘要：
传统的因式分解法只能求出Liénard型常微分方程的特解。我们建议修改的方法，可用于构建一般的解决方案。我们首先证明了我们的方法的有效性，通过处理一个可解形式的修改的Emden型方程，并随后使用它来获得孤立波解的KdV，mKdV，Rosenau-Hyman（RH）和NLS方程。mKdV方程的解，通过所谓的Muira变换，导致KdV方程的奇异解，除了著名的孤子解决方案支持it. We获得的解的不可积RH方程的雅可比函数，并表明，虽然强大的，它是结构上不同的KdV孤子。我们还发现孤子解的NLS方程，占复杂的信封在光学介质中的演变。
英文摘要：
The traditional method of factorization can be used to obtain only the particular solutions of the Liénard type ordinary differential equations. We suggest a modification of the approach that can be used to construct general solutions . We first demonstrate the effectiveness of our method by dealing with a solvable form of the modified Emden-type equation and subsequently employ it to obtain the solitary wave solutions of the KdV, mKdV, Rosenau-Hyman (RH) and NLS equations. The solution of the mKdV equation, via the so-called Muira transform, leads to a singular solution of the KdV equation in addition to the well-known soliton solution supported by it. We obtain the solution of the non-integrable RH equation in terms of the Jacobi function and show that, although robust, it is structurally different from the KdV soliton. We also find the soliton solution of the NLS equation that accounts for the evolution of complex envelopes in an optical medium.
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