Exact periodic wave solutions for some nonlinear partial differential equations

El-Wakil, S.A.; Elgarayhi, A.; Elhanbaly, A.

doi:10.1016/j.chaos.2005.08.063

Cited by 13 publications

(7 citation statements)

References 10 publications

(5 reference statements)

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“…It is of prime importance to investigate exact periodic solutions in terms of elliptic functions, which may lead to breather wave solutions and trigonometric function solutions in limiting cases of nonlinear wave equations in mathematical physics . Recently, some transformation mechanisms were developed to seek periodic solutions of nonlinear wave equations in terms of the Weierstrass elliptic function [58][59][60], four theta functions [58,59,61] and Jacobian elliptic functions [55][56][57]. A uniform method to find all the solutions of nonlinear wave equations does not exist.…”

Section: Breather-like Proton Transportmentioning

confidence: 99%

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Breather-like protonic tunneling in a discrete hydrogen bonded chain with heavy-ionic interactions

Kavitha¹,

Parasuraman²,

Venkatesh³

et al. 2013

Phys. Scr.

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We consider the tunneling motion of protons within a hydrogen bonded (HB) chain. Each proton is subjected to a coulomb interaction from the nearest heavy ions, as well as from the two neighboring protons. We investigate the nonlinear proton dynamics of a HB chain in the semiclassical limit using the coherent state method combined with the Holstein–Primakoff bosonic representation. The protonic transport mechanism has arisen due to the neighboring proton–proton interaction and coherent tunneling of protons along hydrogen bonds and/or around heavy atoms. We construct the exact periodic solutions in terms of elliptic functions by invoking a discrete Jacobian elliptic function method. We present a detailed analysis of the effect of the interaction strength of neighboring protons in the process of bioenergy localization in the form of coherent localized breather modes in a discrete HB chain. A linear stability analysis is performed and the eigenvalues are strictly lying in the imaginary axis, confirming the stable nature of the obtained solutions.

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Section: Breather-like Proton Transportmentioning

confidence: 99%

“…A uniform method to find all the solutions of nonlinear wave equations does not exist. The periodic wave solutions in terms of the Jacobi elliptic functions for the nonlinear PDEs have attracted considerable interest [57] because of the elegant properties of the elliptic functions.…”

Section: Breather-like Proton Transportmentioning

confidence: 99%

Breather-like protonic tunneling in a discrete hydrogen bonded chain with heavy-ionic interactions

Kavitha¹,

Parasuraman²,

Venkatesh³

et al. 2013

Phys. Scr.

View full text Add to dashboard Cite

show abstract

“…In the past decades, there has been significant progression in the development of methods such as the inverse scattering method [1], Hirota's bilinear method [2], similarity transformation method [3,4], homogeneous balance method [5], the sine-cosine method [6], tanh function method [7,8], mapping method [9,10], F-expansion method [11], Riccati equation rational expansion method [12], Jacobi and Weierstrass elliptic function method [13,14] and new generalized Jacobi elliptic function expansion method [15]. In [16][17][18][19], Wang and Chen present a new elliptic function rational expansion method and is more powerful than exiting Jacobi elliptic function method [20] to uniformly construct more new doubly periodic solutions in terms of rational formal Jacobi elliptic function of nonlinear evolution equations.…”

Section: Introductionmentioning

confidence: 99%

New generalized Jacobi elliptic function expansion method

El-Sabbagh

Ali

2008

Communications in Nonlinear Science and Numerical Simulation

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“…Another example is where cnoidal waves appear in shallow water waves, although this is an extremely scarce phenomena. Some interesting communications dealing with the shock wave solutions are found in [1][2][3].…”

Section: Introductionmentioning

confidence: 99%

Shock Wave Solutions for Some Nonlinear Flow Models Arising in the Study of a Non-Newtonian Third Grade Fluid

Aziz

Moitsheki

Fatima

et al. 2013

Mathematical Problems in Engineering

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This study is based upon constructing a new class of closed-form shock wave solutions for some nonlinear problems arising in the study of a third grade fluid model. The Lie symmetry reduction technique has been employed to reduce the governing nonlinear partial differential equations into nonlinear ordinary differential equations. The reduced equations are then solved analytically, and the shock wave solutions are constructed. The conditions on the physical parameters of the flow problems also fall out naturally in the process of the derivation of the solutions.

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Exact periodic wave solutions for some nonlinear partial differential equations

Cited by 13 publications

References 10 publications

Breather-like protonic tunneling in a discrete hydrogen bonded chain with heavy-ionic interactions

Breather-like protonic tunneling in a discrete hydrogen bonded chain with heavy-ionic interactions

New generalized Jacobi elliptic function expansion method

Shock Wave Solutions for Some Nonlinear Flow Models Arising in the Study of a Non-Newtonian Third Grade Fluid

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