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I want to solve the problem: find some non-trivial particular solution of nonlinear PDE. Are the any methods for this? I understand that there is no general method to find general solution, but.. One of the equation is $$u_{t}+u^2 u_{x}-u_{xxx}=0$$

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    $\begingroup$ Not $u_{tt} -u_{xx} = 2u^3 -u$ anymore? I am writing an answer using perturbation method for that. $\endgroup$
    – Shuhao Cao
    Commented Jun 22, 2013 at 6:30

2 Answers 2

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$$u_{t}+u^2 u_y-u_{yyy}=0$$ This PDE may be rewritten as a (modified) Korteweg–de Vries Equation after the substitutions $\displaystyle u(t,y)=\sqrt{6}\,i\,w(t,x)\;$ and $\;y=-x\,$ that give : $$w_t+w_{xxx}+6w^2w_x=0$$ with different kinds of solutions provided in the EqWorld link.

Polyanin's EqWorld that should be recommended for any difficult ODE or PDE !

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You can always look for travelling wave solutions. Setting $u(x,t)=v(x-ct)$ (where $c$ is an arbitrary constant parameter) we find that $v(\xi)$ satisfies an ordinary differential equation: $$v'''=(v^2-c)v'.\tag{1}$$ This can be integrated in quadratures:

  • First, we obviously have $$v''=\frac13v^3-cv+A\tag{2}$$

  • Multiplying (2) by $2v'$, we integrate once more: $$ (v')^2=\frac{1}{6}v^4-cv^2+2Av+B\tag{3}$$

  • Finally, (3) is separable: $$\int\frac{dv}{\sqrt{\frac{1}{6}v^4-cv^2+2Av+B}}=\pm\xi+C.\tag{4}$$

Here $A$, $B$, $C$ are three arbitrary constants of integration. The integral on the left can be computed explicitly in terms of elliptic functions. It simplifies to trigonometric functions if we set $A=B=0$: this is the so-called one-soliton solution of mKdV. For non-zero $A$, $B$ (4) is the simplest example of finite-gap solutions.

Maybe it is worth mentioning that such travelling wave solutions will exist for (more or less) any evolutionary PDE which does not contain $x$, $t$ explicitly. Now as pointed by Raymond Manzoni, your PDE is special: it is an integrable equation which has many special features like infinite number of conserved quantities, an associated linear problem, etc. In particular, this manifests itself in the possibility of constructing multi-soliton solutions.

In case you would like to learn more about integrable stuff, I strongly recommend the book Solitons: Differential equations, symmetries, and infinite-dimensional algebras by M. Jimbo, T. Miwa and E. Date. With a minimal effort, it can be found online.

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