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Substitution of the stationary ansatz into the focusing nonlinear Schrodinger equation gives the real ordinary differential equationMultiply by $2f^{\prime }$ and integrate to obtain the first integralThe finite L2 norm forces the relevant orbit to approach $(f,f^{\prime })=(0,0)$ and hence $C=0$. To justify this rather than assume pointwise decay, the conserved expression first bounds $f$ and $f^{\prime }$, since $f^4+2E{f}^2$ tends to infinity with $|f|$. Thus $f^2$ is uniformly continuous and integrable, and tends to zero at both ends. If $C>0$, then $|f^{\prime }|$ would eventually be bounded away from zero with fixed sign, contradicting $f\to 0$; $C<0$ is incompatible with $(f^{\prime }{)}^2=C-2E{f}^2-f^4$ near zero. Therefore $C=0$ and $f^{\prime }\to 0$.

For $E\ge 0$, the identity $(f^{\prime }{)}^2=-2E{f}^2-f^4$ permits only $f=0$. For a nonzero localized profile put $\kappa =\sqrt{-2E}>0$. ThenA nonzero solution cannot cross zero, because at a zero it also has $f^{\prime }=0$, and uniqueness of the ordinary differential equation would make it identically zero. Its sign is fixed, its maximum absolute value is $\kappa$, and separation on either side of that maximum givesDirect differentiation verifies the profile and its squared L2 norm is $\int f^{2}dx=2\kappa$. The two signs can be incorporated into an arbitrary constant phase of the complex bright soliton of the focusing nonlinear Schrödinger equation.

For a Galilean boost of a nonlinear Schrödinger soliton, define ${\psi }_u(x,t)=e^{i(ux-u^{2}t/2)}{\psi }_0(x-ut,t)$. Differentiating shows that the terms proportional to $u{\partial }_x{\psi }_0$ cancel between $i{\partial }_t$ and $-{\partial }_x^2/2$, as do the added $u^2/2$ terms. The cubic nonlinearity is unchanged because the multiplying phase has modulus one. Hence the boosted field solves the same equation. In particular,whose centre follows $X(t)=ut+X_0$.

For the slowly varying external potential, use the collective-coordinate effective Lagrangian method: retain the localized free profile but let its centre, boost and overall phase vary with time, with its norm fixed. Substitute this finite-parameter ansatz into the field Lagrangian, integrate over space and derive the Euler-Lagrange equations for the parameters. The potential contribution is the external potential averaged over the soliton's density; if it varies little across the profile, it is approximately its value at the centre times the conserved norm. This gives approximate particle-like centre motion while neglecting radiation and profile deformation. No solution of the full perturbed field equation is required for this approximation.

For the harmonic trap, keep the assumed trapped stationary profile $f(y)$, and put $y=x-X(t)$. Take $\Theta (x,t)=v(t)x+\gamma (t)-Et$. Substituting $e^{i\Theta }f(y)$, the imaginary terms are $-i\dot{X}{f}^{\prime }$ on the left and $-iv{f}^{\prime }$ on the right, so choose $v=\dot{X}$. Using the profile equation, the remaining real equation isIt is satisfied providedSince $d(X\dot{X})/dt={\dot{X}}^2-X^2$ for this oscillator, an explicit phase and trajectory areThus harmonic-trap translation of a nonlinear Schrödinger soliton is exact: the trapped profile translates without changing its shape, even though its centre accelerates.