Solving Non-Linear PDEs

We start with a scalar field that is sinusoidal in x and y.
Then we let a PDE decide what happens next, changing every 57.6seconds:
du/dt=u*du/dy-d²u/dy²-d²u/dx²,

d²u/dt²=u*du/dx-d²u/dy²-d²u/dx²,

du/dt=u*du/dy+d³u/dy³-d²u/dy²-d²u/dx²

duk²/dt²=u*du/dy+d³u/dy³-d²u/dy²-d²u/dx²
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