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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.6. Higher-Order Equations
\dfrac{\partial w}{\partial t} + 
\dfrac{\partial ^5 w}{\partial x^5 } + 
5w\dfrac{\partial ^3 w}{\partial x^3} + 
5\dfrac{\partial w}{\partial x}\dfrac{\partial ^2 w}{\partial x^2 } + 
5 w^2\dfrac{\partial w}{\partial x} = 0.
Sawada-Kotera equation.
w_{1,2}(x,t)&=&a-\frac{3R}{1\pm \cos \left( \sqrt{R}\left(
  x-(5a^{2}-5aR+R^{2})t\right) \right) },\\
w_{3,4}(x,t)&=&a-\frac{3R}{1\pm \sin \left( \sqrt{R}\left(
  x-(5a^{2}-5aR+R^{2})t\right) \right) },\\
w_{5,6}(x,t)&=&a+\frac{3R}{1\pm \cosh \left( \sqrt{R}\left(
  x-(5a^{2}+5aR+R^{2})t\right) \right) },\\
w_{7}(x,t)&=&-R\left( 1-\frac{6}{1+\cosh \left( \sqrt{R}\left( x-R^{2}t\right)
  \right) }\right),\\
w_{8}(x,t)&=&R\left( 1-\frac{6}{1-\cos \left( \sqrt{R}\left( x-R^{2}t\right)
  \right) }\right),\\
w_{9,10}(x,t)&=&R\left( 1-\frac{6}{1\pm \sin \left( \sqrt{R}\left(
  x-R^{2}t\right) \right) }\right),\\
w_{11}(x,t)&=&-R \left(1+3 \text{csch}^2\left(\frac{1} {2} \sqrt{R}
  \left(x-R^2 t\right)\right)\right),\\
w_{12}(x,t)&=&a-6R\sec ^{2}\left( \sqrt{R}\left(
  x-(5a^{2}-20aR+16R^{2})t\right) \right),\\
w_{13}(x,t)&=&a+6R\,\text{sech}^{2}\left( \sqrt{R}\left(
  x-(5a^{2}+20aR+16R^{2})t\right) \right),\\
w_{14}(x,t)&=&a-6R\csc ^{2}\left( \sqrt{R}\left(
  x-(5a^{2}-20aR+16R^{2})t\right) \right),\\
w_{15}(x,t)&=&-4R\left( 1-3\,\text{sech}^{2}\left( \sqrt{R}\left(
  x-16R^{2}t\right) \right) \right),\\
w_{16}(x,t)&=&4R\left( 1-3\sec ^{2}\left( \sqrt{R}\left( x-16R^{2}t\right)
  \right) \right),\\
w_{17}(x,t)&=&-4R\csc ^{2}\left( \sqrt{R}\left( x-16R^{2}t\right) \right)
  \left( 2+\cos ^{2}\left( \sqrt{R}\left( x-16R^{2}t\right) \right) \right),\\
w_{18}(x,t)&=&\frac{R\left( \mu ^{2}-5-4\sin \left( \sqrt{R}\left(
  x-R^{2}t\right) \right) \mu -\cos ^{2}\left( \sqrt{R}\left( x-R^{2}t\right)
  \right) \right) }{\left( \mu +\sin \left( \sqrt{R}\left( x-R^{2}t\right)
  \right) \right) ^{2}},\\
w_{19}(x,t)&=&-\frac{R\left( \mu ^{2}-6-4\cosh \left( \sqrt{R}\left(
  x-R^{2}t\right) \right) \mu +\cosh ^{2}\left( \sqrt{R}\left( x-R^{2}t\right)
  \right) \right) }{\left( \mu +\cosh \left( \sqrt{R}\left( x-R^{2}t\right)
  \right) \right) ^{2}},\\
w_{20}(x,t)&=&\frac{R\left( \mu ^{2}-6-4\cos \left( \sqrt{R}\left(
x-R^{2}t\right) \right) \mu +\cos ^{2}\left( \sqrt{R}\left( x-R^{2}t\right)
\right) \right) }{\left( \mu +\cos \left( \sqrt{R}\left( x-R^{2}t\right)
\right) \right) ^{2}}.
Remarks:Solutions given here were obtained by the Projective Riccati equation method with the aid of a computer.
Novelty:New solution(s) / integral(s)
References: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:Thu 22 May 2008 19:11
Edits by author:0

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