The two-dimensional cubic nonlinear Schrödinger equation is used to describe the propagation of an intense laser beam through a medium with Kerr nonlinearity. The coupled two-dimensional cubic nonlinear Schrödinger equations are used to describe interaction of electromagnetic waves with different polarizations in nonlinear optics. In this paper, we solve the above equations by imposing a quadratic condition on the related argument functions and using their symmetry transformations. More complete families of exact solutions of such type are obtained. Many known interesting solutions, such soliton ones, turn out to be special cases of our solutions.
An exactly solvable position-dependent mass Schrödinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical w…
We consider the dynamics of the 3D N-body Schrödinger equation in the presence of a quadratic trap. We assume the pair interaction potential is N^{3β-1}V(N^{β}x). We justify the mean-field approximation and offer a rigorous derivation of the 3D cubic NLS with a quadratic trap. We establish the space-time bound conjectured by Klainerman and Machedon [30] for β in (0,2/7] by adapting and simplifying an argument in Chen and Pavlović [7] which solves the problem for β in (0,1/4) in the absence of a trap.
We consider the 2d and 3d many body Schrödinger equations in the presence of anisotropic switchable quadratic traps. We extend and improve the collapsing estimates in Klainerman-Machedon [24] and Kirkpatrick-Schlein-Staffilani [23]. Together with an anisotropic version of the generalized lens transform in Carles [3], we derive rigorously the cubic NLS with anisotropic switchable quadratic traps in 2d through a modified Elgart-Erdös-Schlein-Yau procedure. For the 3d case, we establish the uniqueness of the corresponding Gross-Pitaevskii hierarchy without the assumption of factorized initial dat…
The Davey-Stewartson equations are used to describe the long time evolution of a three-dimensional packets of surface waves. Assuming that the argument functions are quadratic in spacial variables, we find in this paper various exact solutions modulo the most known symmetry transformations for the Davey-Stewartson equations.
This study presents the solutions of Schrödinger equation for the Non-Central Generalized Inverse Quadratic Yukawa Potential within the framework of Nikiforov-Uvarov. The radial and angular part of the Schrödinger equation are obtained using the method of variable separation. More so, the bound states energy eigenvalues and corresponding eigenfunctions are obtained analytically. Numerical results were obtained for the Generalized Inverse Quadratic Yukawa Potential for comparison sake. It was found out that our results agree with existing literature.
In this work we study the initial boundary value problem associated with the coupled Schrödinger equations {with quadratic nonlinearities, that appears in nonlinear optics}, on the half-line. We obtain local well-posedness for data {in Sobolev spaces} with low regularity, by using a forcing problem on the full line with a presence of a forcing term in order to apply the Fourier restriction method of Bourgain. The crucial point in this work is the new bilinear estimates on the classical Bourgain spaces $X^{s,b}$ with $b<\frac12$, jointly with bilinear estimates in adapted Bourgain spaces that w…
In this work we study the persistence in time of superoscillations for the Schrödinger equation with quadratic time-dependent Hamiltonians. We have solved explicitly the Cauchy initial value problem with three different kind of oscillatory initial data. In order to prove the persistence of superoscillations we have defined explicitly an operator in terms of solutions of a Riccati system associated with the variable coefficients of the Hamiltonian. The operator is defined on a space of entire functions. Particular examples include Caldirola-Kanai and degenerate parametric harmonic oscillator Ha…
We discuss a procedure to determine finite sets $\mathcal{M}$ within the commutant of an algebraic Hamiltonian in the enveloping algebra of a Lie algebra $\mathfrak{g}$ such that their generators define a quadratic algebra. Although independent from any realization of Lie algebras by differential operators, the method is partially based on an analytical approach, and uses the coadjoint representation of the Lie algebra $\mathfrak{g}$. The procedure, valid for non-semisimple algebras, is tested for the centrally extended Schrödinger algebras $\widehat{S}(n)$ for various different choices of alg…
In this paper we propose a modified Lie-type spectral splitting approximation where the external potential is of quadratic type. It is proved that we can approximate the solution to a one-dimensional nonlinear Schroedinger equation by solving the linear problem and treating the nonlinear term separately, with a rigorous estimate of the remainder term. Furthermore, we show by means of numerical experiments that such a modified approximation is more efficient than the standard one.
An effective equation describes a weakly nonlinear wave field evolution governed by nonlinear dispersive PDEs \emph{via} the set of its resonances in an arbitrary big but finite domain in the Fourier space. We consider the Schrödinger equation with quadratic nonlinearity including small external random forcing/dissipation. An effective equation is deduced explicitly for each case of monomial quadratic nonlinearities $ u^2, \, \bar{u}u, \, \bar{u}^2$ and the sets of resonance clusters are studied. In particular, we demonstrate that the nonlinearity $\bar{u}^2$ generates no 3-wave resonances and…
We review different properties related to the Cauchy problem for the (nonlinear) Schrodinger equation with a smooth potential. For energy-subcritical nonlinearities and at most quadratic potentials, we investigate the necessary decay in space in order for the Cauchy problem to be locally (and globally) well-posed. The characterization of the minimal decay is different in the case of super-quadratic potentials.
During recent years, exact solutions of position-dependent mass Schrödinger equations have inspired intense research activities, based on the use of point canonical transformations, Lie algebraic methods or supersymmetric quantum mechanical techniques. Here we highlight the interest of another approach to such problems, relying on quadratic algebras. We illustrate this point by constructing spectrum generating algebras for a class of $d$-dimensional radial harmonic oscillators with $d\ge2$ (including the one-dimensional oscillator on the line via some minor changes) and a specific mass choice.…
In this article, we first employ the concentration compactness techniques to prove existence and stability results of standing waves for nonlinear fractional Schrödinger-Choquard equation \[ i\partial_tΨ+ (-Δ)^αΨ= a |Ψ|^{s-2}Ψ+λ\left( \frac{1}{|x|^{N-β}} \star |Ψ|^p \right)|Ψ|^{p-2}Ψ \ \ \mathrm{in}\ \mathbb{R}^{N+1}, \] where $N\geq 2$, $α\in (0,1)$, $β\in (0, N)$, $s\in (2, 2+\frac{4α}{N})$, $p\in [2, 1+\frac{2α+β}{N})$, and the constants $a, λ$ are nonnegative satisfying $a+λ> 0.$ We then extend the arguments to establish similar results for coupled standing waves of nonlinear fractional Sc…
We study the Cauchy problem for Schrodinger equations with repulsive quadratic potential and power-like nonlinearity. The local problem is well-posed in the same space as that used when a confining harmonic potential is involved. For a defocusing nonlinearity, it is globally well-posed, and a scattering theory is available, with no long range effect for any superlinear nonlinearity. When the nonlinearity is focusing, we prove that choosing the harmonic potential sufficiently strong prevents blow-up in finite time. Thanks to quadratic potentials, we provide a method to anticipate, delay, or pre…
We discuss proofs of nonlocality based on a generalization by Erwin Schrödinger of the argument of Einstein, Podolsky and Rosen. These proofs do not appeal in any way to Bell's inequalities. Indeed, one striking feature of the proofs is that they can be used to establish nonlocality solely on the basis of suitably robust perfect correlations. First we explain that Schrödinger's argument shows that locality and the perfect correlations between measurements of observables on spatially separated systems implies the existence of a non-contextual value-map for quantum observables; non-contextual me…
Some symmetries of time-dependent Schrödinger equations for inverse quadratic, linear, and quadratic potentials have been systematically examined by using a method suitable to the problem. Especially, the symmetry group for the case of the linear potential turns out to be a semi-direct product $SL(2,R) x T_2(R)$ of the $SL(2,R)$ with a two-dimensional real translation group $T_2(R)$. Here, the time variable $t$ transforms as $t \to t^\prime = (ct+d)/(at+b)$ for real constants $a, b, c$, and $d$ satisfying $bc - ad =1$ with an accompanying transformation for the space coordinate $x$.
In this article, we determine the wave front sets of solutions to time dependent Schrödinger equations with a sub-quadratic potential by using the representation of the Schrdingier evolution operator via wave packet transform (short time Fourier transform).
In this paper we study the derivation of a certain type of NLS from many-body interactions of bosonic particles. We consider a model with a finite linear combination of $n$-body interactions, where $n \geq 2$ is an integer. We show that the $k$-particle marginal density of the BBGKY hierarchy converges when particle number goes to infinity, and the limit solves a corresponding infinite Gross-Pitaevskii hierarchy. We prove the uniqueness of factorized solution to the Gross-Pitaevskii hierarchy based on a priori space time estimates. The convergence is established by adapting the arguments origi…
We consider the Schrödinger equation $(i\partial_t+Δ)u=0$ on an $n$-dimensional simplex with Dirichlet boundary conditions. We use a commutator argument along with integration by parts to obtain an observability asymptotic for any one face of the simplex. Rather than the typical observability inequality, we are able to do better as we instead prove a large-time asymptotic.
The dynamics of Gaussian states for open quantum systems described by Lindblad equations can be solved analytically for systems with quadratic Hamiltonians and linear Lindbladians, showing the familiar phenomena of dissipation and decoherence. It is well known that the Lindblad dynamics can be expressed as an ensemble average over stochastic pure-state dynamics, which can be interpreted as individual experimental implementations, where the form of the stochastic dynamics depends on the measurement setup. Here we consider quantum-jump and stochastic Schrödinger dynamics for initially Gaussian s…
The classical quantization of a family of a quadratic Liénard-type equation (Liénard II equation) is achieved by a quantization scheme (M.~C. Nucci. {\em Theor. Math. Phys.}, 168:994--1001, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. This method straightforwardly yields the Schrödinger equation as given in (A.~Ghose~Choudhury and Partha Guha. {\em J. Phys. A: Math. Theor.}, 46:165202, 2013).
After introducing the formalism of the general space and time fractional Schrödinger equation, we concentrate on the time fractional Schrödinger equation and present new results via the elegant language of Fox's H-functions. We show that the general time dependent part of the wave function for the separable solutions of the time fractional Schrödinger equation is the Mittag-Leffler function with an imaginary argument by two different methods. After separating the Mittag-Leffler function into its real and imaginary parts, in contrast to existing works, we show that the total probability is less…
We study the action of metaplectic operators on Wiener amalgam spaces, giving upper bounds for their norms. As an application, we obtain new fixed-time estimates in these spaces for Schrödinger equations with general quadratic Hamiltonians and Strichartz estimates for the Schrödinger equation with potentials $V(x)=\pm|x|^2$.
For a large class of Schrodinger operators, we introduce the hyperbolic quadratic pencils by making the coupling constant dependent on the energy in the very special way. For these pencils, many problems of scattering theory are easier to study. Then, we give some applications to the original Schrodinger operators.
For non-self-adjoint almost-periodic Schrödinger operators, a criterion is given to guarantee that they have both the same spectrum and same Lyapunov exponents with the discrete free Laplacian. As a byproduct, we show that the Moser-Pöschel argument for opening gaps may not be valid for non-self-adjoint operators.
We study the defocusing nonlinear Schrödinger equation in the quarter plane with asymptotically periodic boundary values. By studying an associated Riemann-Hilbert problem and employing nonlinear steepest descent arguments, we construct solutions in a sector close to the boundary whose leading behaviour is described by a single exponential plane wave. Furthermore, we compute the subleading terms in the long time asymptotics of the constructed solutions.
The results from J. Stat. Phys. 159:668-712 & 163:1350-1393, on a quadratic kinetic equation in the analysis of the long time asymptotics of weak turbulence theory for the nonlinear Schrödinger equation, are summarized and placed in context. Additionally, two conjectures on self-similar solutions are presented, and backed with consistency analysis and numerics.
We study the many-body problem of charged particles interacting with their self-generated electromagnetic field. We model the dynamics of the particles by the many-body Maxwell-Schrödinger system, where the particles are treated quantum mechanically and the electromagnetic field is a classical quantity. We prove the existence of a unique local in time solution to this nonlinear initial value problem using a contraction mapping argument.
