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Traveling waves in a cold plasma are governed by $$ \begin{gathered} \frac{\partial p}{\partial t}+\frac{\partial}{\partial x}(\beta u)=0 \\ \frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+E=0 \\ \frac{\partial E}{\partial x}=\omega_{p}{ }^{2}(1-\rho) \end{gathered} $$ Let \(\rho=1+O(c), u=O(c)\), and \(E=O(6)\). Use the method of averaging to determine the temporal as well as the spatial variation of the amplitude and phase of a monochromatic traveling wave.

Consider the equation $$ u_{t t}-c^{2} u_{x x}+u=e u^{3} $$ (a) Write down the Lagrangian corresponding to this equation, (b) determine a first-order expansion for traveling waves with constant wave number and frequency but both spatially and temporally varying amplitude and phase.