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Exact Cylindrical Soliton Solutions of the Sine-Gordon Equation, the Sinh-Gordon Equation and the Periodic Toda Equation
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In this article, the series expansions of the Bessel functions were used to obtain the exact analytic solutions of the sine-Gordon equation, the sinh Gordon equation and the periodic Toda equation corresponding to cylindrical solitons.Abstract:
We consider the sine-Gordon equation, the sinh-Gordon equation and the periodic Toda equation which are written respectively as Δ u +sin u =0, Δ u +sinh u =0 and Δ u n -exp (- u n + u n -1 )+exp (- u n +1 + u n )=0, ( u n = u n + N ' ), where Δ ≡∂ 2 /∂ x 2 +∂ 2 /∂ y 2 . The exact analytic solutions of the above equations corresponding to the cylindrical solitons have been obtained. The solutions are expressed by the series expansions of the Bessel functions.read more
Citations
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Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity
TL;DR: In this article, a traveling-wave solution of the class of equations ∑ p = 1 n 1 α p ∂ p Q ∂ t p + ∑ q = 1 N 2 β q ∂ q Q ∆ x q + ∆ m = 1 M μ m Q m = 0 where α p, β q and μ m are parameters.
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Modified method of simplest equation: Powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs
TL;DR: In this paper, the Taylor series was used to obtain exact traveling-wave solutions of the generalized Rayleigh equation and the extended tanh-equation for the cases when the equations of Bernoulli and Riccati are used as simplest equations.
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Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics
TL;DR: In this paper, the traveling-wave solutions of two classes of equations, namely reaction-diffusion and reaction-telegraph, were obtained by the modified method of simplest equation for the cases when the simplest equation is the equation of Bernoulli or Riccati.
Journal ArticleDOI
Modified method of simplest equation and its application to nonlinear PDEs
TL;DR: The obtained general results are illustrated by obtaining exact solutions of versions of the generalized Kuramoto–Sivashinsky equation, reaction–diffusion equation with density-dependent diffusion, and the reaction-telegraph equation.
Journal ArticleDOI
A Bilinear N-Soliton Formula for the KP Equation
TL;DR: In this paper, the authors considered the bilinear N -soliton solutions for the KP (two dimensional KdV) equation which utilize the Jacobi formula of the matrix algebra.
References
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A new form of Backlund transformations and its relation to the inverse scattering problem
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Solution by the spectral-transform method of a nonlinear evolution equation including as a special case the cylindrical KdV equation
TL;DR: In this article, the authors discuss on the difficulty of different classes of classes of non-equivalence classes of a class of equivalence and the cost of some clumsiness.
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Exact Bessel Type Solution of the Two-Dimensional Toda Lattice Equation
TL;DR: In this paper, the authors have shown that the general superposition of ordinary solitons and the various ripplons is possible for 2-dimensional Toda lattice equations.
Journal ArticleDOI
Return effect for rotationally symmetric solitary wave solutions to the sine-Gordon equation
TL;DR: In this article, rotationally symmetric solitary wave solutions to the sine-Gordon equation (ring waves) are shown to reach a maximum extension and then shrink, which is referred to as return effect.
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