Partial Differential Equation with Travelling Wave Solutions

Question

Let $f: mathbb{R} rightarrow mathbb{R}$ be a smooth function. Consider the partial differential equation:
$$
frac{partial u}{partial t}+frac{partial f(u(x))}{partial x}+frac{partial^{3} u}{partial x^{3}}=0
$$
I wish to find the solutions of the form $u(x, t)=phi(x-c t)$ (traveling wave solutions) where $phi$ is a smooth function. Then I am asked to integrate the obtained ODE, any help would be much appreciated since I am completely lost especially in the substitution process.

GAUSS1860 · Accepted Answer

We first set $gamma=x-alpha t$. Then we obtain the ordinary differential
equation
$$
-alpha frac{d phi}{d gamma}+f(phi(gamma))frac{d phi}{d gamma}+frac{d^{3} phi}{d gamma^{3}}=0.
$$
Integrating once we arrive at
$$
-alpha phi+f(phi(gamma))+frac{d^{2} phi}{d gamma^{2}}=C_{1}
$$
where $C_{1}$ is a constant of integration. One more integration yields
$$
frac{1}{2}left(frac{d phi}{d gamma}right)^{2}=C_{2}+C_{1} phi+frac{alpha}{2} phi^{2}-F(phi(gamma)), quad F(phi)=int_{0}^{phi} f(y) d y
$$
Where $C_{2}$ is another constant of integration.

Partial Differential Equation with Travelling Wave Solutions

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