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**APM346--Misc / perturbed NLS**

« **on:**February 19, 2019, 06:57:24 AM »

Consider the perturbed NLS

\begin{equation}\partial_t q=-\partial_x^2 q-2|q|^2q+\epsilon R\qquad R=R(q,-q^*).\label{1}\end{equation}

It is suggested that we consider also the conjugated equation at the same time

\begin{equation}\partial_t r=\partial_x^2 r+2|r|^2r-\epsilon R^*\qquad r=-q^*.\label{2}\end{equation}

It seems to me that whenever $q$ solves \eqref{1}, $r$ solves \eqref{2}.Then how does \eqref{2} help? What puzzles me more is that when we do the 1st order perturbation theory and collect coefficients according to $\epsilon^1$, it is suggested to write a coupled system of PDE involving $q,r$. Why is this better than just considering one equation, say \eqref{1} alone?

Afterwards we also consider the spectrum of that coupled linear operator, but is that why?

\begin{equation}\partial_t q=-\partial_x^2 q-2|q|^2q+\epsilon R\qquad R=R(q,-q^*).\label{1}\end{equation}

It is suggested that we consider also the conjugated equation at the same time

\begin{equation}\partial_t r=\partial_x^2 r+2|r|^2r-\epsilon R^*\qquad r=-q^*.\label{2}\end{equation}

It seems to me that whenever $q$ solves \eqref{1}, $r$ solves \eqref{2}.Then how does \eqref{2} help? What puzzles me more is that when we do the 1st order perturbation theory and collect coefficients according to $\epsilon^1$, it is suggested to write a coupled system of PDE involving $q,r$. Why is this better than just considering one equation, say \eqref{1} alone?

Afterwards we also consider the spectrum of that coupled linear operator, but is that why?