Exact soliton solutions for the interaction of few-cycle-pulse with nonlinear medium

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640013 ◽  
Author(s):  
Bang-Xing Guo ◽  
Ji Lin
Keyword(s):  
Nonlinear Medium ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Expansion Method ◽  
Periodic Wave ◽  
Interaction Solution ◽  
Coupled Equations ◽  
Waves Interaction ◽  
The One ◽  
Cycle Pulse

We study the Panilevé property of the coupled equations describing the interaction of few-cycle-pulse with nonlinear medium. And we use the consistent tanh expansion (CTE) method to search for exact interaction soliton solutions of the coupled equations. Many interaction solutions are obtained, such as the one kink-one periodic wave interaction solution, one kink-two periodic waves interaction solution, one kink-one dipole soliton interaction solution, one kink-two dipole solitons interaction solution, and one kink-soliton-one periodic wave interaction solution. We also obtain the kink–kink interaction by using Painlevé truncated expansion method.

scholarly journals Investigation of Interaction Solutions for Modified Korteweg-de Vries Equation by Consistent Riccati Expansion Method

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Jin-Fu Liang ◽  
Xun Wang
Keyword(s):  
Vries Equation ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Expansion Method ◽  
Periodic Wave ◽  
Mkdv Equation ◽  
Jacobi Elliptic Function ◽  
Interaction Solutions ◽  
De Vries ◽  
Consistent Riccati Expansion

A consistent Riccati expansion (CRE) method is proposed for obtaining interaction solutions to the modified Korteweg-de Vries (mKdV) equation. Using the CRE method, it is shown that interaction solutions such as the soliton-tangent (or soliton-cotangent) wave cannot be constructed for the mKdV equation. More importantly, exact soliton-cnoidal periodic wave interaction solutions are presented. While soliton-cnoidal interaction solutions were found to degenerate to special resonant soliton solutions for the values of modulus (n) closer to one (upper bound of modulus) in the Jacobi elliptic function, a normal kink-shaped soliton was observed for values of n closer to zero (lower bound).

N-soliton solutions and localized wave interaction solutions of a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyamaf equation

2021 ◽  
pp. 2150277
Author(s):  
Hongcai Ma ◽  
Qiaoxin Cheng ◽  
Aiping Deng
Keyword(s):  
Dynamic Behavior ◽  
Soliton Solution ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Bilinear Transformation ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Ytsf Equation ◽  
Line Soliton

[Formula: see text]-soliton solutions are derived for a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation by using bilinear transformation. Some local waves such as period soliton, line soliton, lump soliton and their interaction are constructed by selecting specific parameters on the multi-soliton solutions. By selecting special constraints on the two soliton solutions, period and lump soliton solution can be obtained; three solitons can reduce to the interaction solution between period soliton and line soliton or lump soliton and line soliton under special parameters; the interaction solution among period soliton and two line solitons, or the interaction solution for two period solitons or two lump solitons via taking specific constraints from four soliton solutions. Finally, some images of the results are drawn, and their dynamic behavior is analyzed.

Nonlocal Symmetry and Consistent Tanh Expansion Method for the Coupled Integrable Dispersionless Equation

2016 ◽  
Vol 71 (3) ◽  
pp. 235-240 ◽  
Author(s):  
Hengchun Hu ◽  
Xiao Hu ◽  
Bao-Feng Feng
Keyword(s):  
Soliton Solutions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Cnoidal Wave ◽  
Nonlocal Symmetry ◽  
Nonlocal Symmetries ◽  
Periodic Wave Solutions ◽  
Wave Solutions

AbstractNonlocal symmetries are obtained for the coupled integrable dispersionless (CID) equation. The CID equation is proved to be consistent, tanh-expansion solvable. New, exact interaction excitations such as soliton–cnoidal wave solutions, soliton–periodic wave solutions, and multiple resonant soliton solutions are discussed analytically and shown graphically.

Modulation instability analysis and optical solitons of the generalized model for description of propagation pulses in optical fiber with four non-linear terms

2020 ◽  
pp. 2150112
Author(s):  
S. U. Rehman ◽  
Aly R. Seadawy ◽  
M. Younis ◽  
S. T. R. Rizvi ◽  
T. A. Sulaiman ◽  
...  
Keyword(s):  
Optical Fibers ◽  
Modulation Instability ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Jacobi Elliptic Function ◽  
Arbitrary Power ◽  
Non Linear ◽  
Complex Phenomena ◽  
Singular Soliton

In this article, we investigate the optical soiltons and other solutions to Kudryashov’s equation (KE) that describe the propagation of pulses in optical fibers with four non-linear terms. Non-linear Schrodinger equation with a non-linearity depending on an arbitrary power is the base of this equation. Different kinds of solutions like optical dark, bright, singular soliton solutions, hyperbolic, rational, trigonometric function, as well as Jacobi elliptic function (JEF) solutions are obtained. The strategy that is used to extract the dynamics of soliton is known as [Formula: see text]-model expansion method. Singular periodic wave solutions are recovered and the constraint conditions, which provide the guarantee to the soliton solutions are also reported. Moreover, modulation instability (MI) analysis of the governing equation is also discussed. By selecting the appropriate choices of the parameters, 3D, 2D, and contour graphs and gain spectrum for the MI analysis are sketched. The obtained outcomes show that the applied method is concise, direct, elementary, and can be imposed in more complex phenomena with the assistant of symbolic computations.

scholarly journals Time-Space Fractional Coupled Generalized Zakharov-Kuznetsov Equations Set for Rossby Solitary Waves in Two-Layer Fluids

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 41 ◽  
Author(s):  
Lei Fu ◽  
Yaodeng Chen ◽  
Hongwei Yang
Keyword(s):  
Rossby Waves ◽  
Expansion Method ◽  
Weakly Nonlinear ◽  
Multi Scale ◽  
Coupled Equations ◽  
Time Space ◽  
Weakly Nonlinear Waves ◽  
Waves Interaction ◽  
Geostrophic Vortex ◽  
First Time

In this paper, the theoretical model of Rossby waves in two-layer fluids is studied. A single quasi-geostrophic vortex equation is used to derive various models of Rossby waves in a one-layer fluid in previous research. In order to explore the propagation and interaction of Rossby waves in two-layer fluids, from the classical quasi-geodesic vortex equations, by employing the multi-scale analysis and turbulence method, we derived a new (2+1)-dimensional coupled equations set, namely the generalized Zakharov-Kuznetsov(gZK) equations set. The gZK equations set is an extension of a single ZK equation; they can describe two kinds of weakly nonlinear waves interaction by multiple coupling terms. Then, for the first time, based on the semi-inverse method and the variational method, a new fractional-order model which is the time-space fractional coupled gZK equations set is derived successfully, which is greatly different from the single fractional equation. Finally, group solutions of the time-space fractional coupled gZK equations set are obtained with the help of the improved ( G ′ / G ) -expansion method.

Periodic-wave solutions and asymptotic properties for a (3+1)-dimensional generalized breaking soliton equation in fluids and plasmas

2021 ◽  
pp. 2150344
Author(s):  
Rui-Dong Chen ◽  
Yi-Tian Gao ◽  
Xin Yu ◽  
Ting-Ting Jia ◽  
Gao-Fu Deng ◽  
...  
Keyword(s):  
Asymptotic Properties ◽  
Theta Functions ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Soliton Equation ◽  
One Dimensional ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Riemann Theta Functions ◽  
The One

In this paper, a (3+1)-dimensional generalized breaking soliton equation is investigated. Based on the one- and two-dimensional Riemann theta functions, one- and two-periodic-wave solutions are derived. We observe that the one-periodic wave is one-dimensional and is viewed as a superposition of the overlapping waves, placed one period apart. With certain parameters, the symmetric feature appears in the two-periodic wave, and the two-periodic wave degenerates to the one-periodic wave. With the series expansions, we explore the relations between the soliton and periodic-wave solutions. According to those relations, asymptotic properties for the periodic-wave solutions to approach to the soliton solutions under certain amplitude conditions are derived.

scholarly journals Some New Exact Solutions of (1+2)-Dimensional Sine-Gordon Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Wei-Xiong Chen ◽  
Ji Lin
Keyword(s):  
Gordon Equation ◽  
Direct Method ◽  
Wave Interaction ◽  
Expansion Method ◽  
Periodic Wave ◽  
Sine Gordon Equation ◽  
Interaction Solutions ◽  
New Exact Solutions ◽  
Propagation Properties ◽  
A Direct Method

We use a generalized tanh function expansion method and a direct method to study the analytical solutions of the (1+2)-dimensional sine Gordon (2DsG) equation. We obtain some new interaction solutions among solitary waves and periodic waves, such as the kink-periodic wave interaction solution, two-periodic solitoff solution, and two-toothed-solitoff solution. We also investigate the propagation properties of these solutions.

Breather-wave, periodic-wave and traveling-wave solutions for a (2 + 1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation for an incompressible fluid

2021 ◽  
pp. 2150261
Author(s):  
Yuan Shen ◽  
Bo Tian ◽  
Chen-Rong Zhang ◽  
He-Yuan Tian ◽  
Shao-Hua Liu
Keyword(s):  
Incompressible Fluid ◽  
Theta Function ◽  
Traveling Wave ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Polynomial Expansion Method

In this paper, the investigation is conducted on a (2 + 1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation for an incompressible fluid. Via the Riemann theta function, periodic-wave solutions are derived, and breather-wave solutions are constructed with the aid of the extended homoclinic test approach. Based on the polynomial expansion method, several traveling-wave solutions are derived. Besides, we observe that the amplitude of the breather keeps unchanged during the propagation and the traveling wave which is kink shaped propagates stably. Furthermore, we analyze the transition between the periodic-wave and soliton solutions, which implies that the periodic-wave solutions tend to the soliton solutions via a limiting procedure.

scholarly journals Two Reliable Methods for Solving the (3 + 1)-Dimensional Space-Time Fractional Jimbo-Miwa Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-30 ◽  
Author(s):  
Sekson Sirisubtawee ◽  
Sanoe Koonprasert ◽  
Chaowanee Khaopant ◽  
Wanassanun Porka
Keyword(s):  
Exact Solutions ◽  
Dimensional Space ◽  
Soliton Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Space Time ◽  
The Novel ◽  
Wave Solutions ◽  
Liouville Derivative

We investigate methods for obtaining exact solutions of the (3 + 1)-dimensional nonlinear space-time fractional Jimbo-Miwa equation in the sense of the modified Riemann-Liouville derivative. The methods employed to analytically solve the equation are the G′/G,1/G-expansion method and the novel G′/G-expansion method. To the best of our knowledge, there are no researchers who have applied these methods to obtain exact solutions of the equation. The application of the methods is simple, elegant, efficient, and trustworthy. In particular, applying the novel G′/G-expansion method to the equation, we obtain more exact solutions than using other existing methods such as the G′/G-expansion method and the exp-Φ(ξ)-expansion method. The exact solutions of the equation, obtained using the two methods, can be categorized in terms of hyperbolic, trigonometric, and rational functions. Some of the results obtained by the two methods are new and reported here for the first time. In addition, the obtained exact explicit solutions of the equation characterize many physical meanings such as soliton solitary wave solutions, periodic wave solutions, and singular multiple-soliton solutions.

On the classification of conservation laws and soliton solutions of the long short-wave interaction system

2018 ◽  
Vol 32 (18) ◽  
pp. 1850202 ◽  
Author(s):  
Mustafa Inc ◽  
Aliyu Isa Aliyu ◽  
Abdullahi Yusuf ◽  
Dumitru Baleanu
Keyword(s):  
Conservation Laws ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Expansion Method ◽  
Short Wave ◽  
Short Transverse ◽  
Interaction System ◽  
Conservation Theorem

In this paper, the classification of conservation laws (Cls) of the long short-wave interaction system (LSWS) which appears in fluid mechanics as well as plasma physics is implemented using two Cls theorems, namely, the multipliers approach and the new conservation theorem. The LSWS describes the interaction between one long longitudinal wave and one short transverse wave propagating in a generalized elastic medium. The zeroth-order multipliers and the nonlinear self-adjoint substitutions of the model are derived. Considering the fact that the new conservation theorem needs Lie point symmetries in constructing Cls, we derive the point symmetries of a system of nonlinear partial differential equations (NPDEs) acquired by transforming the model into real and imaginary components. Moreover, we derive some kink-type, bell-shaped, singular and combined soliton solutions to the model using the powerful sine-Gordon expansion method (SGEM). Some figures are presented to show the physical interpretations of the acquired results.

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