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Nikolay K. Vitanov Zlatinka I. Dimitrova Kaloyan N. Vitanov 《Entropy (Basel, Switzerland)》2021,23(1)
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 ; (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 to solution for arbitrary finite values of . 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. 相似文献
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Zhiber-Shabat方程的孤立波解与周期波解 总被引:1,自引:1,他引:0
结合齐次平衡法原理并利用F展开法,再次研究了Zhiber-Shabat方程的各种椭圆函数周期解.当椭圆函数的模m分别趋于1或0时,利用这些椭圆函数周期解,得到了Zhiber-Shabat方程的各种孤子解和三角函数周期解,从而丰富了相关文献中关于Zhiber-Shabat波方程的解的类型. 相似文献
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We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and 0, the hyperbolic function solutions (including the solitary wave solutions) and trigonometric solutions are also given respectively. 相似文献
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M. A. Abdou 《Nonlinear dynamics》2008,52(3):277-288
The improved F-expansion method with a computerized symbolic computation is used to construct the exact traveling wave solutions
of four nonlinear evolution equations in physics. As a result, many exact traveling wave solutions are obtained which include
new soliton-like solutions, trigonometric function solutions, and rational solutions. The method is straightforward and concise,
and it holds promise for many applications. 相似文献
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CHEN Jiang YANG Kong-Qing HE Hong-Sheng 《理论物理通讯》2007,48(5):877-880
A new generalized F-expansion method is introduced and applied to the study of the (2+1)-dimensional Boussinesq equation. The further extension of the method is discussed at the end of this paper. 相似文献
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CHEN Jiang HE Hong-Sheng YANG Kong-Qing 《理论物理通讯》2005,44(2):307-310
A generalized F-expansion method is introduced and applied to (3+1 )-dimensional Kadomstev-Petviashvili(KP) equation. As a result, some new Jacobi elliptic function solutions of the equation are found, from which the trigonometric function solutions and the solitary wave solutions can be obtained. The method can also be extended to other types of nonlinear evolution equations in mathematical physics. 相似文献
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The Periodic Wave Solutions for Two Nonlinear Evolution Equations 总被引:14,自引:0,他引:14
ZHANGJin-Liang WANGMing-Liang CHENGDong-Ming FANGZong-De 《理论物理通讯》2003,40(2):129-132
By using the F-expansion method proposed recently, the periodic wave solutions expressed by Jacobi elliptic functions for two nonlinear evolution equations are derived. In the limit cases, the solitary wave solutions and the other type of traveling wave solutions for the system are obtained. 相似文献
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本文针对耦合Schrodinger-Boussinesq方程组,借助于F-展开法得到了用不同Jacobi椭圆函数表示的一系列周期波解.在极限情况下,还求出了对应的孤立波解. 相似文献
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We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. By using extended F-expansion method, many
periodic wave solutions expressed by various Jacobi elliptic functions for the Klein-Gordon-Schrödinger equations are obtained. In the limit cases, the solitary wave solutions and
trigonometric function solutions for the equations are also
obtained. 相似文献