AMT C27/25

KJELL Theory

We are concerned with a partial differential equation.

\begin{eqnarray*}

\frac{\partial U}{\partial t} &=& I U + G U^2 - H UV + \Phi(U) + J\left(\cos \frac{\beta z}{h} -1\right) U \\

\frac{\partial V}{\partial t} &=& {\cal h}(U^2-V) + \mu {\cal H}(U)

\end{eqnarray*}

where $\Phi$ and ${\cal H}$ are linear operators invariant under congruences of the two-dimensional space. In effect they are functions of $\nabla^2$.

As an approximation we replace the second equation by one of form.

\begin{eqnarray*}

V &=& \Lambda(U^2)

\end{eqnarray*}

Suppose now that it is assumed that $U$ and $V$ are periodic in $z$ with period $2\pi h$ and in $x$ with period $2\pi\rho$. Then

\begin{eqnarray*}

U &=& \sum A_{m,n} e^{i\left(\frac{m x}{\rho}+ \frac{n z}{h}\right)} \\

V &=& \sum D_{m,n} e^{i\left(\frac{m x}{\rho}+ \frac{n z}{h}\right)}

\end{eqnarray*}

If $\frac{\partial U}{\partial t}=\sum K_{m,n} e^{i\left(\frac{m x}{\rho}+ \frac{n z}{h}\right)}$ and if \begin{eqnarray*} \Phi(U) &=& \sum \phi \left(\frac{m^2}{\rho^2}+\frac{n^2}{h^2}\right) A_{m,n} e^{i\left(\frac{m x}{\rho}+ \frac{n z}{h}\right)} \\

\Lambda(U) &=& \sum \lambda \left(\frac{m^2}{\rho^2}+\frac{n^2}{h^2}\right) A_{m,n} e^{i\left(\frac{m x}{\rho}+ \frac{n z}{h}\right)}

\end{eqnarray*}

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