Discussions on diffraction and the dispersion for traveling wave solutions of the (2+1)-dimensional paraxial wave equation

Durur, Hülya; Yokuş, Asıf

doi:10.1007/s40096-021-00419-z

Cited by 16 publications

(3 citation statements)

References 43 publications

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“…Nonlinear evolution equations represent many physical applications and dynamic processes [1], which play a significant role in the fields of nonlinear science such as fluid mechanics, fiber optics and ocean dynamics [2][3][4][5]. In recent years, the exact solutions of nonlinear evolution equations such as solitons, breathers and lump waves have been studied extensively by the Hirota bilinear approach [6][7][8][9][10], analytical methods [11][12][13][14], symbolic computation using the neural network method [15][16][17][18], Bäcklund transformations [19,20] and Darboux transformations [21][22][23][24]. Solitons keep their shapes and velocities unchanged in the processes of propagation [25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

The mixed solutions and nonlinear wave transitions for the (2 + 1)-dimensional Sawada-Kotera equation

Bi¹,

Guo²

2022

Phys. Scr.

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In this paper, we will obtain the mixed solutions of the 2 + 1 -dimensional Sawada-Kotera equation by the Hirota bilinear method and the long wave limit method. Through applying the long wave limit method to the N-soliton solutions, we will obtain the corresponding lump solutions. Besides, we will get the mixed solutions of lump, breather and soliton solutions. Under the transition condition, we will obtain different types of nonlinear waves which include quasi-anti-dark soliton, multi-peak soliton, M-shaped soliton, quasi-periodic soliton and W-shaped soliton from the breath wave and the lump solutions. In particular, we will derive the inelastic collision modes, the elastic collision modes and the collision-free modes under specific conditions. Finally, we will display the mixed solutions and the nonlinear waves through illustrations.

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Section: Introductionmentioning

confidence: 99%

The mixed solutions and nonlinear wave transitions for the (2 + 1)-dimensional Sawada-Kotera equation

Bi¹,

Guo²

2022

Phys. Scr.

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“…1/ ' G -expansion method, finite difference method and Laplace perturbation method [11], modified sub-equation method [12], modified Kudryashov methods [13].…”

Section: Introductionmentioning

confidence: 99%

Soliton solutions for perturbed Radhakrishnan-Kundu-Lakshmanan equation

Demiray

BAYRAKCI

Yildirim

2022

Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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To find some soliton solutions of the equation, the perturbed Radhakrishnan-Kundu-Lakshmanan (RKL) equation has been considered. For this purpose, GKM (generalized Kudryashov method), which is one of the solution methods of nonlinear evolution equations (NLEEs), has been applied to the perturbed RKL equation. First, considered the nonlinear partial differential equation, is reduced to an ordinary differential equation with the help of the traveling wave transformation. Afterward, obtained the algebraic equation system through the balance principle was solved with the help of Wolfram Mathematica 12. Thus, some new soliton solutions of the discussed equation have been obtained. Both 2D and 3D graphics have been drawn with the help of Wolfram Mathematica 12 by giving some values to obtained these new solutions.

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“…So the discussion of NPDEs exact solutions in the nonlinear sciences is very important. Over the past few years, many researchers have used this beneficial method extensively, for example, the Jacobi elliptic expansion method [1], modified Kudryashov method [2,3], the tanh method [4], sub-equation analytical method [5], the inverse scattering method [6], the first integral method [7,8], the extended tanh-function method [9,10], the Hirota's direct method [11], the auxiliary equation method [12], improved Bernoulli sub-equation function method [13], expansion method [14], (G /G, 1/G)-expansion method [15,16,17], generalized exponential rational function method [18,19,24], Sinh-Gordon function method [20], Sine-Gordon expansion method [21], Bernoulli sub-equation method [22], (G /G)-expansion method [23]. The equation of the Eckhaus is as follows [25]:…”

Section: Introductionmentioning

confidence: 99%

Construction of Different Types of Traveling Wave Solutions of the Relativistic Wave Equation Associated with the Schrödinger Equation

Yokuş¹

2021

mmnsa

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In this study, an alternative method has been applied to obtain the new wave solution of mathematical equations used in physics, engineering, and many applied sciences. We argue that this method can be used for some special nonlinear partial differential equations (NPDEs) in which the balancing methods are not integer. A number of new complex hyperbolic trigonometric traveling wave solutions have been successfully generated in the Eckhaus equation (EE) and nonlinear Klein-Gordon (nKG) equation models associated with the Schrödinger equation. The graphs representing the stationary wave are presented by giving specific values to the parameters contained in these solutions. Finally, some discussions about new complex solutions are given. It is discussed by giving physical meaning to the constants in traveling wave solutions, which are physically important as well as mathematically. These discussions are supported by three-dimensional simulation. In order to eliminate the complexity of the process and to save time, computer package programs have been utilized.

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Discussions on diffraction and the dispersion for traveling wave solutions of the (2+1)-dimensional paraxial wave equation

Cited by 16 publications

References 43 publications

The mixed solutions and nonlinear wave transitions for the (2 + 1)-dimensional Sawada-Kotera equation

The mixed solutions and nonlinear wave transitions for the (2 + 1)-dimensional Sawada-Kotera equation

Soliton solutions for perturbed Radhakrishnan-Kundu-Lakshmanan equation

Construction of Different Types of Traveling Wave Solutions of the Relativistic Wave Equation Associated with the Schrödinger Equation

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