scholarly journals New Solutions for the Generalized BBM Equation in terms of Jacobi and Weierstrass Elliptic Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Alvaro H. Salas ◽  
Lorenzo J. Martinez H ◽  
David L. Ocampo R
Keyword(s):  
Elliptic Function ◽  
Function Method ◽  
Soliton Solutions ◽  
Elliptic Functions ◽  
Jacobi Elliptic Function ◽  
Series Method ◽  
Power Series Method ◽  
Weierstrass Elliptic Function ◽  
Generalized Bbm Equation ◽  
Bbm Equation

The Jacobi elliptic function method is applied to solve the generalized Benjamin-Bona-Mahony equation (BBM). Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered. A power series method is also applied in some particular cases. Some solutions are expressed in terms of the Weierstrass elliptic function.

Study on soliton solutions of the longitudinal wave equation and magneto-electro-elastic circular rod dynamical model

2021 ◽  
Vol 35 (13) ◽  
pp. 2150168
Author(s):  
Adel Darwish ◽  
Aly R. Seadawy ◽  
Hamdy M. Ahmed ◽  
A. L. Elbably ◽  
Mohammed F. Shehab ◽  
...  
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Longitudinal Wave ◽  
Physical Interpretation ◽  
Elliptic Function ◽  
Function Method ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Jacobi Elliptic Function ◽  
Two Dimensional

In this paper, we use the improved modified extended tanh-function method to obtain exact solutions for the nonlinear longitudinal wave equation in magneto-electro-elastic circular rod. With the aid of this method, we get many exact solutions like bright and singular solitons, rational, singular periodic, hyperbolic, Jacobi elliptic function and exponential solutions. Moreover, the two-dimensional and the three-dimensional graphs of some solutions are plotted for knowing the physical interpretation.

Dispersive soliton solutions for the Salerno equation for the nonlinear discrete electrical lattice in the forbidden bandgaps

2021 ◽  
Author(s):  
Ahmed M. Elsherbeny ◽  
Reda El-Barkouky ◽  
Aly R. Seadawy ◽  
Hamdy M. Ahmed ◽  
Rabab M. I. El-Hassani ◽  
...  
Keyword(s):  
Periodic Solutions ◽  
Elliptic Function ◽  
Function Method ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Jacobi Elliptic Function ◽  
Two Dimensional ◽  
Bright Solitons ◽  
Jacobi Elliptic Function Method

In this paper, the modified Jacobi elliptic function method is applied for Salerno equation which describes the nonlinear discrete electrical lattice in the forbidden bandgaps. Dark and bright solitons are obtained. Also, periodic solutions and periodic Jacobi elliptic function solutions are reported. Moreover, for the physical illustration of the obtained solutions, three-dimensional and two-dimensional graphs are presented.

scholarly journals Exact Solutions for the Generalized BBM Equation with Variable Coefficients

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Cesar A. Gómez ◽  
Alvaro H. Salas
Keyword(s):  
Exact Solutions ◽  
Function Method ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Iteration Algorithm ◽  
Variational Iteration ◽  
Generalized Bbm Equation ◽  
Bbm Equation

The variational iteration algorithm combined with the exp-function method is suggested to solve the generalized Benjamin-Bona-Mahony equation (BBM) with variable coefficients. Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered.

Quasi-Periodic, Periodic Waves, and Soliton Solutions for the Combined KdV-mKdV Equation

2009 ◽  
Vol 64 (9-10) ◽  
pp. 639-645 ◽  
Author(s):  
Emad A.-B. Abdel-Salam
Keyword(s):  
Elliptic Function ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Travelling Wave ◽  
Mkdv Equation ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions ◽  
De Vries ◽  
Jacobi Elliptic Function Method

By introducing the generalized Jacobi elliptic function, a new improved Jacobi elliptic function method is used to construct the exact travelling wave solutions of the nonlinear partial differential equations in a unified way. With the help of the improved Jacobi elliptic function method and symbolic computation, some new exact solutions of the combined Korteweg-de Vries-modified Korteweg-de Vries (KdV-mKdV) equation are obtained. Based on the derived solution, we investigate the evolution of doubly periodic and solitons in the background waves. Also, their structures are further discussed graphically.

scholarly journals Stability and Extraction of New Optical Wave Solutions of Cascaded System in (2 + 1)-dimensions

2021 ◽  
Author(s):  
Hitender Kumar ◽  
Parveen Parveen ◽  
Sunita Dahiya ◽  
Anand Kumar ◽  
Manjeet Singh Gautam
Keyword(s):  
Elliptic Function ◽  
Data Transfer ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Dynamical Behaviour ◽  
Auxiliary Equation ◽  
Jacobi Elliptic Function ◽  
Optical Wave ◽  
Auxiliary Equation Method ◽  
Weierstrass Elliptic Function

Abstract This paper uses the new modified sub-ODE method, the new extended auxiliary equation method, and the new Jacobi elliptic function expansion method to revisit the (2+1)-dimensional coupled nonlinear Schr¨odinger equation with cascading effect. As a consequence, dark, bright, kinkantikink, singular solitons, Weierstrass elliptic function, doubly periodic, and complex optical soliton solutions are retrieved. All solutions are described, along with the existence criterion on the parameters. As solitons are used for data transfer, the obtained results may be found usage in optical couplers, birefringed fibres, and optoelectronic devices. A comparison of the obtained results with those found in the literature is given. The dynamical behaviour of some of the obtained solutions has been explored for suitable choices of the parameters. Using the property of Hamiltonian systems, the solitons stability is determined.

scholarly journals Bright, dark, and singular optical soliton solutions for perturbed Gerdjikov-Ivanov equation

2021 ◽  
Vol 25 (Spec. issue 2) ◽  
pp. 151-156
Author(s):  
Esma Ulutas ◽  
Mustafa Inc ◽  
Dumitru Baleanu ◽  
Sunil Kumar
Keyword(s):  
Exact Solutions ◽  
Optical Fibers ◽  
Elliptic Function ◽  
Soliton Solutions ◽  
Elliptic Functions ◽  
Optical Soliton ◽  
Jacobi Elliptic Function ◽  
Jacobi Elliptic Functions ◽  
Ansatz Method ◽  
Pulse Dynamics

This study consider Gerdjikov-Ivanov equation where the perturbation terms appear with full non-linearity. The Jacobi elliptic function ansatz method is implemented to obtain exact solutions of this equation that models pulse dynamics in optical fibers. It is retrieved some bright, dark optical and singular solitons profile in the limiting cases of the Jacobi elliptic functions. The constraint conditions depending on the parameters for the existence of solitons are also presented.

scholarly journals Compactons and topological solitons of the Drinfel'd–Sokolov system

2014 ◽  
Vol 19 (2) ◽  
pp. 209-224
Author(s):  
Mustafa Inc ◽  
Eda Fendoglu ◽  
Houria Triki ◽  
Anjan Biswas
Keyword(s):  
Elliptic Function ◽  
Function Method ◽  
Soliton Solutions ◽  
Ansatz Method ◽  
Topological Soliton ◽  
Topological Solitons ◽  
Wave Solutions

This paper presents the Drinfel’d–Sokolov system (shortly D(m, n)) in a detailed fashion. The Jacobi’s elliptic function method is employed to extract the cnoidal and snoidal wave solutions. The compacton and solitary pattern solutions are also retrieved. The ansatz method is applied to extract the topological 1-soliton solutions of the D(m, n) with generalized evolution. There are a couple of constraint conditions that will fall out in order to exist the topological soliton solutions.

Analytical Solutions of Two Space-Time Fractional Nonlinear Models Using Jacobi Elliptic Function Expansion Method

2021 ◽  
pp. 173-188
Author(s):  
Zillur Rahman ◽  
M. Zulfikar Ali ◽  
Harun-Or-Roshid ◽  
Mohammad Safi Ullah
Keyword(s):  
Elliptic Function ◽  
Nonlinear Models ◽  
Applied Mathematics ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Space Time ◽  
Jacobi Elliptic Function ◽  
Function Expansion ◽  
Partial Models ◽  
Jacobi Elliptic Function Expansion

In this manuscript, the space-time fractional Equal-width (s-tfEW) and the space-time fractional Wazwaz-Benjamin-Bona-Mahony (s-tfWBBM) models have been investigated which are frequently arises in nonlinear optics, solid states, fluid mechanics and shallow water. Jacobi elliptic function expansion integral technique has been used to build more innovative exact solutions of the s-tfEW and s-tfWBBM nonlinear partial models. In this research, fractional beta-derivatives are applied to convert the partial models to ordinary models. Several types of solutions have been derived for the models and performed some new solitary wave phenomena. The derived solutions have been presented in the form of Jacobi elliptic functions initially. Persevering different conditions on a parameter, we have achieved hyperbolic and trigonometric functions solutions from the Jacobi elliptic function solutions. Besides the scientific derivation of the analytical findings, the results have been illustrated graphically for clear identification of the dynamical properties. It is noticeable that the integral scheme is simplest, conventional and convenient in handling many nonlinear models arising in applied mathematics and the applied physics to derive diverse structural precise solutions.

A Jacobi elliptic function method for nonlinear arrays of vortices

2012 ◽  
Vol 86 (12) ◽  
pp. 1107-1113 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. M. Tharwat ◽  
A. Yildirim ◽  
M. A. Abdelkawy
Keyword(s):  
Elliptic Function ◽  
Function Method ◽  
Jacobi Elliptic Function ◽  
Jacobi Elliptic Function Method

Jacobi Elliptic Function Rational Expansion Method with Symbolic Computation to Construct New Doubly-periodic Solutions of Nonlinear Evolution Equations

2004 ◽  
Vol 59 (9) ◽  
pp. 529-536 ◽  
Author(s):  
Yong Chen ◽  
Qi Wang ◽  
Biao Lic
Keyword(s):  
Periodic Solutions ◽  
Symbolic Computation ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Jacobi Elliptic Function ◽  
Rational Form

A new Jacobi elliptic function rational expansion method is presented by means of a new general ansatz and is very powerful, with aid of symbolic computation, to uniformly construct more new exact doubly-periodic solutions in terms of rational form Jacobi elliptic function of nonlinear evolution equations (NLEEs). We choose a (2+1)-dimensional dispersive long wave equation to illustrate the method. As a result, we obtain the solutions found by most existing Jacobi elliptic function expansion methods and find other new and more general solutions at the same time. When the modulus of the Jacobi elliptic functions m→1 or 0, the corresponding solitary wave solutions and trigonometric function (singly periodic) solutions are also found.

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