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Let a real localized profile satisfy $Ef=-f^{\prime \prime }/2-f^3+y^{2}f/2$. 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.

Modulational instability is growth of long-wavelength perturbations of a uniform or periodic wave due to coupling between its modulation and nonlinear dynamics. In the focusing nonlinear Schrodinger equation, the amplitude and phase perturbations can have imaginary modulation frequencies even though the unperturbed wave is an exact solution.

Nonlinear Schrödinger blowup analysis studies whether a solution of the Focusing nonlinear Schrodinger equation remains bounded in its natural Sobolev space or develops an unbounded gradient norm in finite time.

The nonlinear Schrödinger equation adds an amplitude-dependent interaction to Schrödinger-type wave evolution. Its Laplacian term disperses the complex wave amplitude, while the sign and form of the nonlinear term determine focusing or defocusing behavior. A one-dimensional cubic specialization is the focusing nonlinear Schrodinger equation; members of this broader family need not be integrable systems.

Write $r=\sqrt{\mu /\delta }$. At first order the amplitude-phase perturbation is $A=r+\epsilon (A_1+ir\theta )+O({\epsilon }^2)$. Expanding the cubic term in the focusing nonlinear Schrodinger equation gives $A|A{|}^2=r^3+\epsilon (3r^2{A}_1+i{r}^3\theta )+O({\epsilon }^2)$. The constant terms cancel because $\delta r^2=\mu$. Separating real and imaginary parts yieldsFor the Fourier normal modes, these equations give $-i\Omega {\hat{A}}_1=-\chi r{K}^2\hat{\theta }$ and $-ir\Omega \hat{\theta }+(2\mu -\chi K^2){\hat{A}}_1=0$. Their determinant vanishes exactly whenThe focusing nonlinear Schrodinger modulation dispersion is negative for $0<K^2<2\mu /\chi$. One of the two frequencies then has positive imaginary part and grows exponentially: the constant state has modulational instability. The growth rate is $\sqrt{\chi K^2(2\mu -\chi K^2)}$, maximized at $K^2=\mu /\chi$ with value $\mu$. Modes beyond the unstable band have real frequencies. The zero-wavenumber phase mode is neutral, with possible linear phase drift from a uniform amplitude perturbation.

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.

Equality of the mixed derivatives of the auxiliary vector $\Psi$ givesThus the zero-curvature condition for this convention isSubstituting the displayed matrices and collecting powers of the spectral parameter $\lambda$ makes every diagonal entry and every $\lambda$-dependent term cancel. The remaining matrix isConsequently compatibility is equivalent toso the requested constant isThis is the AKNS Lax pair for the nonlinear Schrodinger equation.

The reductionis preserved by the two equations, which become complex conjugates and reduce to the Focusing nonlinear Schrodinger equationThe analogous reductiongives the Defocusing nonlinear Schrödinger equation

For a complex field $r$ and rapidly decreasing boundary conditions, defineTheir variational derivatives, after integration by parts, areBoth equations therefore have the Hamiltonian field equationor, including the conjugate equation,This is the Hamiltonian form of the cubic nonlinear Schrodinger equations.

Now putwhere $E$ and $f$ are real and $f$ is smooth and rapidly decreasing. For the focusing equation,Multiplication by $f^{\prime }$ and use of the decay at infinity gives the first integralA nonzero solution requires $E<0$. Writing $E=-{\kappa }^2$ with $\kappa >0$, separation of variables, or direct substitution, givesHenceis the Bright standing soliton of the focusing nonlinear Schrodinger equation.

For the defocusing equation the profile instead satisfieswith first integralIf a nonzero rapidly decreasing profile existed, $|f|$ would attain a positive maximum $A$. At that point $f^{\prime }=0$, so the identity forces $E=A^2>0$. But along either tail, where $0<|f|<A$, the right-hand side $f^2(f^2-E)$ is negative, which is impossible. Thus there is no nonzero solution of the prescribed form, as recorded by No rapidly decaying standing wave for the defocusing cubic nonlinear Schrodinger equation.