Lecture 18, 18.300, Tue Apr 12, 2022.
Summary
Continue with Gas Dynamics, Acoustics, and Strings.
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#065 Do example: Nonlinear Klein Gordon u_tt - u_xx + V(u) = 0. SEE BELOW.
#057 Shocks in Gas Dynamics.

Nonlinear Klein Gordon  u_tt - u_xx + V(u) = 0.
Write equation in characteristic form using variable v = u_t + u_x
Then:     u_t + u_x = v
          v_t - v_x = -V(u)
--- Note coupling of left and right characteristics.
--- Use dt time step argument to construct solution. If initial data
    is compact support, it stays so.
    Maximum speed of propagation for signals is 1.
--- Recal notions of domain of dependence and domain of influence.
    Note difference with special case of 1D wave equation, where
    only the edges of the domain matter.

Note that, for generic V(u) the equation is semi-linear, and the
solution may cease to exist because singularities form.
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