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In the normalizationthe one-dimensional focusing nonlinear Schrodinger equation admits a localized soliton on zero background. The negative cubic term balances the spreading caused by dispersion. For frequency $E<0$, let $\kappa =\sqrt{2|E|}$. ThenThe constants $x_0$ and ${\theta }_0$ specify the centre and constant phase. The coefficient $1/2$ of the second derivative fixes the relation between frequency, width and amplitude; other common nonlinear Schrödinger equation conventions rescale these quantities.

To check the solution, the hyperbolic secant identity givesHence $-f_E^{\prime \prime }/2-f_E^3=-{\kappa }^2{f}_E/2=E{f}_E$, exactly the stationary equation. The peak amplitude is $\kappa$, the width scale is $1/\kappa$, and ${\int }_{\mathbb{R}}|f_E{|}^{2}dx=2\kappa$, so its L2 norm is finite.

A Galilean boost of a nonlinear Schrödinger soliton produces the moving solutionIts centre travels at constant velocity $u$, while its shape and L2 norm remain fixed. ^[ref]

A dark soliton is a localized dip in the intensity or number density of a nonlinear wave on a nonzero background. The associated complex field changes phase across the dip. This differs from a Bright standing soliton of the focusing nonlinear Schrodinger equation localized on a zero background.

For the defocusing nonlinear Schrödinger equation normalizationa travelling solution with $0\le U<1/\sqrt{2}$ isThe far-field intensity is $|\psi {|}^2\to 1$, while at the centre it is $2U^2$. Thus $U=0$ gives a stationary black soliton with a zero at its centre; $U>0$ gives a shallower grey soliton. These names describe the dip relative to the background, without requiring an optical interpretation of $\psi$.

Substitution checks the balance directly. With $\xi =x-Ut$, ${\psi }^{\prime }=A^2{\mathrm{sech}}^2(A\xi /\sqrt{2})/\sqrt{2}$ and $1-|\psi {|}^2=A^2{\mathrm{sech}}^2(A\xi /\sqrt{2})$. The real part of ${\psi }^{\prime \prime }+(1-|\psi {|}^2)\psi$ cancels, and its imaginary part equals $2iU{\psi }^{\prime }=-2i{\psi }_t$. The width grows like $1/A$ as the dip becomes shallow near the limiting speed.

Writing a defocusing nonlinear Schrödinger field as $\psi =\sqrt{\rho }{e}^{iS}$ turns it into a continuity equation for $\rho$ and a velocity equation for $v=S_x$, including a quantum-pressure term.

In a common normalization, the defocusing case is $i{\psi }_t+{\psi }_{xx}-2|\psi {|}^2\psi =0$. The sign distinguishes it from the focusing nonlinear Schrödinger equation.

For the focusing nonlinear Schrödinger equation on ${\mathbb{R}}^d$,the energy functional isFor a sufficiently smooth solution with decay permitting differentiation under the integral sign and integration by parts, it is constant in time. Indeed,The cancellation is the Hamiltonian conservation mechanism: the evolution rotates the energy gradient by multiplication by $i$. The focusing sign makes the nonlinear contribution negative, so conserved energy alone need not bound the L2 norm of the gradient.^[ref]

For less regular solutions the displayed computation is only formal. To inherit the identity, approximate by smooth solutions and justify convergence of both energy terms. Convergence in Sobolev space $H^1$ controls the nonlinear term through a suitable Sobolev embedding only in the applicable exponent range, or with additional integrability. An arbitrary weak solution must not be assumed to satisfy the identity solely because the smooth equation does.

The focusing cubic nonlinear Schrödinger equation has the normalizationthe positive cubic term focuses the field and permits bright solitons on zero background.

The Gravitational Hartree equation couples a complex wave function to the attractive Newtonian gravitational potential generated by its own squared modulus. In dimension $N\ge 3$, a dimensionless convention iswhere $C_N=(N-2)|S^{N-1}|$ and $|S^{N-1}|$ is the area of the unit sphere. The minus sign in the fundamental solution makes $\varphi \le 0$ for nonnegative density, with the potential selected to decay at infinity when the convolution permits it.

The convolution makes the nonlinearity nonlocal: changing the density at one point changes the potential elsewhere. The Newtonian kernel and its dimension differ from a local cubic nonlinear Schrödinger equation. In three dimensions, the Hardy–Littlewood–Sobolev inequality controls the interaction energy and leads to Global H1 solutions of the three-dimensional gravitational Hartree equation.

The lens transform converts free Schrödinger equation evolution into evolution with a harmonic trapping potential by rescaling space, changing time and adding a quadratic phase. With the convention $i{\partial }_{s}u=-\frac{1}{2}\Delta u$ on ${\mathbb{R}}^d$, putThen $i{\partial }_{t}v=-\frac{1}{2}\Delta v+\frac{1}{2}|x{|}^{2}v$. The prefactor preserves the L2 space norm under the spatial change of variables. Infinite free time $s\to \pm \infty$ is represented by finite endpoints $t\to \pm \pi /2$.^[ref]

For $i{\partial }_{s}u=-\frac{1}{2}\Delta u+\lambda |u{|}^{p-1}u$, the transformed nonlinear coefficient is $\lambda (\cos t{)}^{d(p-1)/2-2}$. It remains constant precisely at $p=1+4/d$. This calculation explains why the transform is particularly useful for the mass-critical nonlinear Schrödinger equation. Its time interval and normalization are essential parts of the formula.

For $i{u}_t+\Delta u+|u{|}^{p-1}u=0$, a smooth real weight $\chi$ defines the localized mass $I_{\chi }(t)=\int \chi |u{|}^2$. For a sufficiently regular decaying solution and a weight with the required integrability,andThese formulas follow by substituting the nonlinear Schrödinger equation and integration by parts. The Hessian controls the gradient contribution; the fourth derivative term records variation of the cutoff.

Choose ${\chi }_R(x)=R^2\chi (x/R)$, with $\chi (x)=|x{|}^2$ near zero and constant outside a larger ball. The interior terms reproduce the virial identity, while the remaining terms are supported in the transition region or controlled by spatial tails. A bounded weight makes $I_{{\chi }_R}$ meaningful for finite mass without requiring finite variance. A blowup proof still needs estimates that control those tail errors.

For the nonlinear Schrödinger equationwith real-valued $F$, a sufficiently regular, decaying solution with finite L2 norm conservesDifferentiating gives $M^{\prime }(t)=2\mathrm{Re}\int u_t\overline{u}$. Substitute $u_t=i\Delta u+iF(|u{|}^2)u$. Integration by parts makes $\int \Delta u\overline{u}=-\int |\nabla u{|}^2$ real, and $\int F(|u{|}^2)|u{|}^2$ is real as well, so multiplication by $i$ gives zero real part.

The real coefficient is essential to this calculation; phase covariance alone does not imply mass conservation. For an $H^1$ solution obtained by a well-posed approximation theory, the invariant extends from smooth solutions by continuity in the L2 norm.^[ref]

Rémi Carles studied critical nonlinear Schrödinger equations with and without a harmonic trapping potential. His work gives a lens transform relating the two evolutions in the mass-critical case. This permits information about existence and concentration in one equation to be transferred to the other, while keeping track of the transformed time interval.^[ref]