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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.6. Higher-Order Equations
\frac{\partial u}{\partial t}+
\omega\frac{\partial^5u}{\partial x^5}+
\alpha u\frac{\partial^3u}{\partial x^3}+
\beta\frac{\partial u}{\partial x}\frac{\partial^2u}{\partial x^2}+
\gamma u^{2}\frac{\partial u}{\partial x}=0.$\newline
General fifth-order KdV equation.
Exact solutions: 
u_{1}(x,t)=\frac{k(A+2\alpha +\beta )}{\gamma }\left( 3\tanh ^{2}\left(
\frac{\sqrt{-k}}{\gamma }\left( \gamma x-2Bk^{2}t\right) \right) -2\right).
u_{2}(x,t)=\frac{k(A+2\alpha +\beta )}{\gamma }\left( 3\coth ^{2}\left(
\frac{\sqrt{-k}}{\gamma }\left( \gamma x-2Bk^{2}t\right) \right) -2\right).
u_{3}(x,t)=-\frac{k(A+2\alpha +\beta )}{\gamma }\left( 3\tan ^{2}\left(
\frac{\sqrt{k}}{\gamma }\left( \gamma x-2Bk^{2}t\right) \right) +2\right).
u_{4}(x,t)=-\frac{k(A+2\alpha +\beta )}{\gamma }\left( 3\cot ^{2}\left(
\frac{\sqrt{k}}{\gamma }\left( \gamma x-2Bk^{2}t\right) \right) +2\right).
u_{5}(x,t)=-\frac{k(A-2\alpha -\beta )}{\gamma }\left( 3\tanh ^{2}\left(
\frac{\sqrt{-k}}{\gamma }\left( \gamma x+2Ck^{2}t\right) \right) -2\right).
u_{6}(x,t)=-\frac{k(A-2\alpha -\beta )}{\gamma }\left( 3\coth ^{2}\left(
\frac{\sqrt{-k}}{\gamma }\left( \gamma x+2Ck^{2}t\right) \right) -2\right).
u_{7}(x,t)=\frac{k(A-2\alpha -\beta )}{\gamma }\left( 3\tan ^{2}\left( \frac{
\sqrt{k}}{\gamma }\left( \gamma x+2Ck^{2}t\right) \right) +2\right).
u_{8}(x,t)=\frac{k(A-2\alpha -\beta )}{\gamma }\left( 3\cot ^{2}\left( \frac{
\sqrt{k}}{\gamma }\left( \gamma x+2Ck^{2}t\right) \right) +2\right).
A=\sqrt{(2\alpha +\beta )^{2}-40\omega \gamma }\text{, \ \ }B=\beta
^{2}+A\beta +2\alpha \beta -12\gamma \omega \text{ \ \ and \ \ }C=-\beta
^{2}+A\beta -2\alpha \beta +12\gamma \omega ,
Some important particular cases are :
\item Kaup-Kupershmidt equation (KK equation) 
\begin{equation}  \label{eq01}
u_t+u_{xxxxx}+10 uu_{xxx}+25u_xu_{xx}+20 u^2u_x=0.
\item Sawada-Kotera equation [3] (SK equation)
\begin{equation}  \label{eq02}
u_t+u_{xxxxx}+5 uu_{xxx}+5 u_xu_{xx}+5 u^2u_x=0.
\item Lax equation 
\begin{equation}  \label{eq04}
u_t+ u_{xxxxx}+\, 10 \,uu_{xxx}+20 u_xu_{xx}+30 u^2u_x=0.
\item Ito equation 
\begin{equation}  \label{eq05}
u_t+ u_{xxxxx}+3 \,uu_{xxx}+6 u_xu_{xx}+2 \,u^2u_x=0.
Remarks:Here, $\alpha $, $\beta $, $\gamma $ and $\omega $ are arbitrary 
parameters with $\gamma \neq 0$.

This equation describes motions of long waves in shallow water under gravity
and in a one-dimensional nonlinear lattice and it is an important
mathematical model with wide applications in quantum mechanics and nonlinear
optics. Typical examples are widely used in various fields such as solid
state physics, plasma physics, fluid physics and quantum field theory. A
great deal of research work has been invested during the past decades for
the study of the fKdV equation. The main goal of these studies was directed
towards its analytical and numerical solutions. Several different
approaches, such as Blackund transformation, a bilinear form, and a Lax
pair, have been used independently by which soliton and multi-soliton
solutions are obtained. Ablowitz et al. [1] implemented the inverse
scattering transform method to handle the nonlinear equations of physical
significance where soliton solutions and rational solutions were developed.

The solutions given here were obtained with the
aid of \emph{Mathematica}  by using the tanh method [2].
Novelty:New solution(s) / integral(s)
References:[1] M. J. Ablowitz, and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.
[2] Malfliet W. Solitary wave solutions of nonlinear wave equations. Am J Phys 1992, 60(7), pp. 650654.
[3] Alvaro H. Salas, Some solutions for a type of generalized Sawada-Kotera equation, Applied Mathematics and Computation, Vol. 196, Issue 2, 2008, pp. 812-817.
Author/Contributor's Details
Last name:Salas
First name:Alvaro
Middle(s) name:Humberto
Affiliation:Universidad de Caldas-Universidad Nacional de Colombia
Statistic information
Submission date:Mon 19 May 2008 01:04
Edits by author:1
Last edit by author:Mon 16 Jun 2008 08:46

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