MiniLogo

EqWorld

The World of Mathematical Equations

IPM Logo

Exact Solutions Methods Software For Authors Math Forums

EqArchive: Add Equation/Solution > View Equation

 English only

View Equation

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

Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.7. Systems of Two Equations
Equation(s):\noindent
$\displaystyle u_t=L[u]+uf(u^2-w^2)+wg(w/u)$,\hfill\break
$\displaystyle w_t=L[w]+wf(u^2-w^2)+ug(w/u)$.\hfill\break
Solution(s),
Transformation(s),
Integral(s)
:
Here, $L$ is an arbitrary linear differential operator with respect to the coordinate
$x$ (of any order in derivatives), whose coefficients can be dependent on $x$.
\medskip
\noindent
Solution:\hfill\break
$$
u=r(x,t)\cosh\varphi(t),\quad w=r(x,t)\sinh\varphi(t),
$$
where the function $\varphi=\varphi(t)$ is determined by the separable first-order ordinary differential equation\hfill\break
$$
\varphi'_t=g(\tanh\varphi),
$$
and the function $r=r(x,t)$ is determined by the differential equation\hfill\break
$$
r_t=L[r]+rf(r^2).\eqno(*)
$$
Remarks:{\it Remark 1.} 
There is a solution of the form
$$
u=r(x,t)\sinh\varphi(t),\quad w=r(x,t)\cosh\varphi(t),
$$

{\it Remark 2.}
In the general case, equation (*) admits an stationary solution $r=r(x)$.
If the $L$ is a constant-coefficient linear differential operator,
then equation (*) admits an exact, traveling-wave solution $r=r(z)$, where
$z=kx-\lambda t$ with arbitrary constants $k$ and~$\lambda$. 

{\it Remark 3.}
The linear differential operator $L$ can be dependent on $x_1,\ldots,x_n$.
Novelty:New equation(s) & solution(s) / integral(s)
Author/Contributor's Details
Last name:Polyanin
First name:Andrei
Country:Russia
City:Moscow
Statistic information
Submission date:Wed 06 Dec 2006 14:40
Edits by author:0

Edit (Only for author/contributor)


The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.

Copyright © 2006-2011 Andrei D. Polyanin, Alexei I. Zhurov and Alexander L. Levitin