Vitanov Nikolay K, Dimitrova Zlatinka I, Vitanov Kaloyan N
Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 4, 1113 Sofia, Bulgaria.
Institute of Solid State Physics, Bulgarian Academy of Sciences, Blvd. Tzarigradsko Chaussee 72, 1784 Sofia, Bulgaria.
Entropy (Basel). 2020 Dec 23;23(1):10. doi: 10.3390/e23010010.
The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a "small" parameter ϵ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of ϵ to solution for arbitrary finite values of ϵ. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.
本文的目的是讨论用于获取非线性偏微分方程精确解的简单方程法（SEsM），并表明几种获取此类方程精确解的著名方法与简单方程法相关。更详细地说，我们表明，对于简单方程法步骤2中特定形式的函数以及指数函数类型的微分方程的简单方程，广田方法与简单方程法的一个特定情况相关。我们通过获得科特韦格 - 德弗里斯方程的三孤子解、非线性薛定谔方程的二孤子解以及铁磁材料自旋动力学的石森方程的孤子解，来说明简单方程法的这个特定情况。然后我们表明，简单方程法的一个特定情况可用于重现伯格斯方程和科特韦格 - 德弗里斯方程情况下的逆散射变换方法的方法。这个特定情况与简单方程法步骤2的一个特定情况的使用相关。这一步骤与以下方面相关：（i）将求解的非线性偏微分方程的解表示为包含“小”参数ϵ的幂次的幂级数展开；（ii）借助傅里叶级数求解由此表示产生的微分方程，以及（iii）从ϵ的小值时获得的解过渡到ϵ的任意有限值时的解。最后，我们表明常用的齐次平衡法、扩展齐次平衡法、辅助方程法、雅可比椭圆函数展开法、F - 展开法、修正简单方程法、试探函数法和第一积分法与简单方程法的特定情况相关。
