Waves/1D Solutions

Template:1DWavesTOC As you have learned in the previous sections, the linear waves you studied in the previous section tend to disperse. However, there are nonlinear waves described by nonlinear partial differential equations (PDE) which admits solutions with nondispersing wave packets.

Here are two primary examples:

Sine-Gordon

This is described by the PDE

$\underset{x}{\overset{2}{\partial }}\varphi -\underset{t}{\overset{2}{\partial }}\varphi =\sin (\varphi )$.

KdV

This is described by the PDE

${\partial }_t\varphi +\underset{x}{\overset{3}{\partial }}\varphi +6\varphi {\partial }_x\varphi =0$.

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