= Harmonic-trap translation of a nonlinear Schrödinger soliton
{title2=$\Theta=\dot X(x-X/2)-Et+\Theta_0,\quad\ddot X+X=0$}
Let a real localized profile satisfy $Ef=-f''/2-f^3+y^2f/2$. Then $\psi(x,t)=e^{i\Theta(x,t)}f(x-X(t))$ solves the cubic <focusing nonlinear Schrodinger equation> in the potential $x^2/2$ with the displayed phase and any harmonic-oscillator trajectory $X$. Imaginary terms require $\Theta_x=\dot X$; real terms require $\ddot X=-X$ and the remaining time-dependent phase. The translation is exact and causes no profile deformation, unlike a generic <collective-coordinate effective Lagrangian> approximation.
Back to article page
