$\frac{d{u}_j}{dt}=6u_j\frac{u_{j+1}-u_{j-1}}{2x}-\frac{u_{j+2}-2u_{j+1}+2u_{j-1}-u_{j-2}}{2x^3}.$

To ensure the solution is stable, the stability condition must be satisfied

$1\mathrm{.}089x^3.$

Write a MATLAB program that solves the equation in the discretization form using RK$4$ above.

Solve it in the region $-10x10$ with a grid size $x=0\mathrm{.}1,$ and use periodic boundary

conditions: $x(-10)=x(10)$

Integrate from $t=0$ to $t=2,$ using an appropriate time$-$step that satisfies the stability

condition. For each of the initial conditions below, plot the solution at $t=2$ and comment on

the results.

Use a single soliton as initial condition, that is$,u(x,0)=u_1(x,0).$ Set $v=20$ and $x_0=0.$

$u_1(x,t)=-\frac{v}{2cos{h}^2(\frac{1}{2}\sqrt[\phantom{2}]{v}(x-vt-x_0))}.$

Two$-$soliton solurion:

u$($x$,0)=-6/$cosh$^2($x$)$