Numerical Method for KdV travelling waves

Question

Can someone please direct me to the best NUMERICAL method or some references for solving $$-cu_x + uu_x + u_{xxx} = 0$$ with periodic boundary conditions. This governs travelling waves of the KdV equation with speed $c$. I assume that I should be using some iteration method, but I'm not sure which will give me non-trivial solutions (solitons, cnoidal waves). Thanks :)

Answer 1

This is not an answer to the question about numerical method. Nevertheless, the result below can be numericaly evaluated in order to compare to the direct numerical methods.

$$-cu_x+uu_x+u_{xxx}=0=(u-c)u_x+u_{xxx}$$ This is an ODE of the autonomous kind. The usual change of function is :

$u_x=f(u) \quad\to\quad u_{xx}=f'f \quad\to\quad u_{xxx}=f''f^2+f'^2f$

$(u-c)u_x+u_{xxx}=0=(u-c)f+f''f^2+f'^2f=0$

$$(u-c)+f''f+f'^2=0=(u-c)+(f'f)'$$ $$(u^2-2cu+c_1)+2f'f=0$$ $$(\frac{1}{3}u^3-cu^2+c_1u+c_2)+f^2=0$$ $$f(u)=\pm\sqrt{-\frac{1}{3}u^3+cu^2-c_1u-c_2}=u_x$$ $$x=\pm\int \frac{du}{\sqrt{-\frac{1}{3}u^3+cu^2-c_1u-c_2}}$$ $x(u)$ is an elliptic function of $u$.

At this point, the constants $c_1$ and $c_2$ have to be determined according to the boundary conditions.

Inverting the function $x(u)$ in order to analytically express $u(x)$ might be more or less complicated, depending on $c_1$ and $c_2$. It might be impossible without convenient special functions. In practice, it isn't worthwhile since all ends with numerical calculus : the direct numerical integration of $\int \frac{du}{\sqrt{-\frac{1}{3}u^3+cu^2-c_1u-c_2}}$ allows all further computations.