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Register a new userExact solutions for the general fifth order KdV equation by the extended tanh method
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Added by
Alvaro Salas Humberto
Publication date
2008
fields
Physics
and research's language is
English
Authors
Alvaro Salas
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In this paper we show some exact solutions for the general fifth order KdV equation. These solutions are obtained by the extended tanh method.
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Regarding $N$-soliton solutions, the trigonometric type, the hyperbolic type, and the exponential type solutions are well studied. While for the elliptic type solution, we know only the one-soliton solution so far. Using the commutative B{a}cklund transformation, we have succeeded in constructing the KdV static elliptic $N$-soliton solution, which means that we have constructed infinitely many solutions for the $wp$-function type differential equation.
Symmetries of a differential equations is one of the most important concepts in theory of differential equations and physics. One of the most prominent equations is KdV (Kortwege-de Vries) equation with application in shallow water theory. In this paper we are going to explain a particular method for finding symmetries of KdV equation, which is called Harrison method. Our tools in this method are Lie derivatives and differential forms, which will be discussed in the first section more precisely. In second chapter we will have some analysis on the solutions of KdV equation and we give a method, which is called first integral method for finding the solutions of KdV equation.
this paper we show some new exact solutions for the generalized modified Degasperis$-$Procesi equation (mDP equation)
begin{abstract} We show that if the initial profile $qleft( xright) $ for the Korteweg-de Vries (KdV) equation is essentially semibounded from below and $int^{infty }x^{5/2}leftvert qleft( xright) rightvert dx<infty,$ (no decay at $-infty$ is required) then the KdV has a unique global classical solution given by a determinant formula. This result is best known to date. end{abstract}
We present a novel computational methodology for solving the scalar nonlinear Helmholtz equation (NLH) that governs the propagation of laser light in Kerr dielectrics. The methodology addresses two well-known challenges in nonlinear optics: Singular behavior of solutions when the scattering in the medium is assumed predominantly forward (paraxial regime), and the presence of discontinuities in the % linear and nonlinear optical properties of the medium. Specifically, we consider a slab of nonlinear material which may be grated in the direction of propagation and which is immersed in a linear medium as a whole. The key components of the methodology are a semi-compact high-order finite-difference scheme that maintains accuracy across the discontinuities and enables sub-wavelength resolution on large domains at a tolerable cost, a nonlocal two-way artificial boundary condition (ABC) that simultaneously facilitates the reflectionless propagation of the outgoing waves and forward propagation of the given incoming waves, and a nonlinear solver based on Newtons method. The proposed methodology combines and substantially extends the capabilities of our previous techniques built for 1Dand for multi-D. It facilitates a direct numerical study of nonparaxial propagation and goes well beyond the approaches in the literature based on the augmented paraxial models. In particular, it provides the first ever evidence that the singularity of the solution indeed disappears in the scalar NLH model that includes the nonparaxial effects. It also enables simulation of the wavelength-width spatial solitons, as well as of the counter-propagating solitons.
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