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The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.5. Third-Order Equations
Equation(s):$$ \dfrac{\partial w}{\partial t} - 
\dfrac{\partial^3 w}{\partial t \partial x^2}+
(b+1) w^2\dfrac{\partial w}{\partial x}
 = b\dfrac{\partial w}{\partial x}\dfrac{\partial^2 w}{\partial x^2}+ 
w\dfrac{\partial^3 w}{\partial x^3}.$$
Generalized modified Degasperis - Procesi equation (mDP equation).
Solution(s),
Transformation(s),
Integral(s)
:
Solutions :\\
$w_1(x,t)=
\dfrac{6k(b+2)}{b+1}\left(\tan
   ^2\left(\dfrac{1}{2}
   \sqrt{k} (2x-\alpha t)\right)+\cot^2\left(\dfrac{1}{2}\sqrt{k} (2x-\alpha
t)\right)\right)-\dfrac{\alpha-8 b
   k-16 k }{2(
   b+1)}.$
$$w_2(x,t)=
-\dfrac{6k(b+2)}{b+1}\left(\tanh
   ^2\left(\dfrac{1}{2}
   \sqrt{-k} (2x-\alpha t)\right)+\coth^2\left(\dfrac{1}{2}\sqrt{-k} (2x-\alpha
t)\right)\right)-\dfrac{\alpha-8 b
   k-16 k }{2(
   b+1)}.$$
$w_3(x,t)=
\dfrac{6k(b+2)}{b+1}\left(\tan
   ^2\left(\dfrac{1}{2}
   \sqrt{k} (2x+\alpha' t )\right)+\cot^2\left(\dfrac{1}{2}\sqrt{k} (2x+\alpha' t)\right)\right)+
   \dfrac{\alpha'+ 8 b k+16 k}{2(b+1)}.$
$$w_4(x,t)=
-\dfrac{6k(b+2)}{b+1}\left(\tanh ^2\left(\dfrac{1}{2}
   \sqrt{-k} (2x+\alpha' t )\right)+\coth^2\left(\dfrac{1}{2}\sqrt{-k} (2x+\alpha' t)\right)\right)+
   \dfrac{\alpha'+ 8 b k+16 k}{2(b+1)}.$$
$w_5(x,t)=
\dfrac{6k(b+2)}{b+1}\tan^2\left(\dfrac{1}{2} \sqrt{k} ( 2x-\beta
t)\right)-\dfrac{\beta-8 b k-16 k}{2
   (b+1)}.$\\
$w_6(x,t)=
-\dfrac{6k(b+2)}{b+1}\tanh^2\left(\dfrac{1}{2} \sqrt{-k} ( 2x-\beta
t)\right)-\dfrac{\beta-8 b k-16 k}{2
   (b+1)}.$\\
$w_7(x,t)=
\dfrac{6k(b+2)}{b+1}\tan^2\left(\dfrac{1}{2}
   \sqrt{k} \left(2x- \beta't\right)\right)-\dfrac{\beta\,'-8bk-16k}{2(b+1)}.$\\
$w_8(x,t)=
-\dfrac{6k(b+2)}{b+1}\tanh^2\left(\dfrac{1}{2}
   \sqrt{-k} \left(2x- \beta\,'t\right)\right)-\dfrac{\beta\,'-8bk-16k}{2(b+1)}.$\\
$w_9(x,t)=
\dfrac{6k(b+2)}{b+1}\cot^2\left(\dfrac{1}{2}
   \sqrt{k} \left(2x- \beta t\right)\right)-\dfrac{\beta-8bk-16k}{2(b+1)}.$\\
$w_{10}(x,t)=
-\dfrac{6k(b+2)}{b+1}\coth^2\left(\dfrac{1}{2}
   \sqrt{-k} \left(2x- \beta t\right)\right)-\dfrac{\beta-8bk-16k}{2(b+1)}.$\\
$w_{11}(x,t)=
\dfrac{6k(b+2)}{b+1}\cot^2\left(\dfrac{1}{2}
   \sqrt{k} \left(2x- \beta't\right)\right)-\dfrac{\beta\,'-8bk-16k}{2(b+1)}.$\\
$w_{12}(x,t)=
-\dfrac{6k(b+2)}{b+1}\coth^2\left(\dfrac{1}{2}
   \sqrt{-k} \left(2x- \beta\,'t\right)\right)-\dfrac{\beta\,'-8bk-16k}{2(b+1)}.$\\
Here, 
$$\alpha=b+1+\sqrt{(b+1)^2-256 b k^2(b+2)},\quad \alpha\,'=-(b+1)+\sqrt{(b+1)^2-256 b k^2(b+2)},$$
$$\beta=b+1+\sqrt{(b+1)^2-16 b k^2 (b+2)},\quad\beta\,'=b+1-\sqrt{(b+1)^2-16 b k^2 (b+2)}.$$
Novelty:New solution(s) / integral(s)
References:Liu Z. and Ouyang Z., A note on solitary waves for modified forms of Camassa–Holm and Degasperis–Procesi equations, Physics Letters A, 2007, 366 (4), pp. 377-381.
Author/Contributor's Details
Last name:Salas
First name:Alvaro
Middle(s) name:Humberto
Country:Colombia
City:Manizales
Affiliation:Universidad Nacional de Colombia, Universidad de Caldas
Statistic information
Submission date:Mon 19 May 2008 14:07
Edits by author:2
Last edit by author:Wed 04 Jun 2008 22:44

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