Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation

Vitanov, Nikolay K.; Dimitrova, Zlatinka I.

doi:10.2478/jtam-2018-0005

Cited by 43 publications

(24 citation statements)

References 26 publications

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“… Step 2. The considered modified version of the SEM (MSEM) expresses the solution of Equation by making an ansatz for v ( ξ ) as

v ((), ξ) = \sum_{i = 0}^{m} A_{i} {(\frac{ϕ^{'} ((), ξ)}{ϕ ((), ξ)})}^{i}, A_{m} \neq 0,

where A i ( i = 0,1, … ,m) are parameters to be determined, and m is a positive integer that can be determined by the balance algorithm as in Step 3 of SEM. ϕ ( ξ ) is an unspecified function to be determined subsequently. Step 3.…”

Section: The Simple Equation Methodsmentioning

confidence: 99%

“…The SEM and its variants have been successfully applied for finding exact solutions of nonlinear Schrödinger equation, coupled Konno‐Oono equations, nonlinear telegraph equation, generalized Davey‐Stewartson and Zakharov equations, dispersive water wave equations, generalized (2 + 1)‐dimensional nonlinear evolution equations (NLEEs), and the (4 + 1)‐dimensional Fokas equation . See also the included bibliography therein.…”

Section: Introductionmentioning

confidence: 99%

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New kink solutions for the van der Waals p‐system

Az-Zo’bi

2019

Math Methods in App Sciences

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Communicated by: Y. Qin MSC Classification: 35C07; 35M30; 35Q35; 47J35; 65Z05; 76Nxx The simple equation method and modified simple equation method are employed to seek exact traveling wave solutions to the (1 + 1)-dimensional van der Waals gas system in the viscosity-capillarity regularization form. Under the help of Mathematica, new classes of kink solutions are derived. Numerical simulations with special choices of the free parameters are displayed by threeand two-dimensional plots. The two methods demonstrate simplicity, reliability, and efficiency. KEYWORDS modified simple equation method, simple equation method, solitary waves, systems of mixed type, traveling wave solutions, van der Waals system

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“… Step 2. The considered modified version of the SEM (MSEM) expresses the solution of Equation by making an ansatz for v ( ξ ) as

v ((), ξ) = \sum_{i = 0}^{m} A_{i} {(\frac{ϕ^{'} ((), ξ)}{ϕ ((), ξ)})}^{i}, A_{m} \neq 0,

Section: The Simple Equation Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New kink solutions for the van der Waals p‐system

Az-Zo’bi

2019

Math Methods in App Sciences

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“…Below we shall apply the modified method of simplest equation. The current version of the methodology used by our research group is based on the possibility of use of more than one simplest equation [65]. We shall describe this version of the methodology and below we shall use the particular case when the solutions of the studied nonlinear PDE are obtained by use of a single simplest equation.…”

Section: The Modified Methods Of Simplest Equationmentioning

confidence: 99%

“…The modified method of simplest equation has shown its effectiveness on the basis of numerous applications, such as obtaining exact traveling wave solutions of generalized Swift -Hohenberg equation and generalized Rayleigh equation [56], generalized Degasperis -Procesi equation and b-equation [57],extended Korteweg-de Vries equations [59,62],generalized Fisher equation, generalized Huxley equation [54], generalized Kuramoto -Sivashinsky equation, reaction -diffusion equation, reaction -telegraph equation [53], etc. [60], [61], [65].…”

Section: Introductionmentioning

confidence: 99%

On the Exact Traveling Wave Solutions of a Hyperbolic Reaction-Diffusion Equation

Jordanov¹,

Vitanov²

2018

Studies in Computational Intelligence

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We discuss a class of hyperbolic reaction-diffusion equations and apply the modified method of simplest equation in order to obtain an exact solution of an equation of this class (namely the equation that contains polynomial nonlinearity of fourth order). We use the equation of Bernoulli as a simplest equation and obtain traveling wave solution of a kink kind for the studied nonlinear reaction-diffusion equation.

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“…Thus we have to substitute the word “simplest” by the word “simple”. A variant of SEsM based on two simple equations was applied in [ 96 ] and the first description of the algorithm was made in [ 71 ] and then in [ 72 , 73 , 74 ]. For more applications of particular cases of the algorithm see [ 97 , 98 ].…”

Section: Introductionmentioning

confidence: 99%

Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods

Vitanov

Dimitrova

Vitanov

2020

Entropy

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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.

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Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation

Cited by 43 publications

References 26 publications

New kink solutions for the van der Waals p‐system

New kink solutions for the van der Waals p‐system

On the Exact Traveling Wave Solutions of a Hyperbolic Reaction-Diffusion Equation

Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods

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