We obtain time dependent $q$-Gaussian wave-packet solutions to a non linear Schrödinger equation recently advanced by Nobre, Rego-Montero and Tsallis (NRT) [Phys. Rev. Lett. 106 (2011) 10601]. The NRT non-linear equation admits plane wave-like solutions ($q$-plane waves) compatible with the celebrated de Broglie relations connecting wave number and frequency, respectively, with energy and momentum. The NRT equation, inspired in the $q$-generalized thermostatistical formalism, is characterized by a parameter $q$, and in the limit $q \to 1$ reduces to the standard, linear Schrödinger equation. The $q$-Gaussian solutions to the NRT equation investigated here admit as a particular instance the previously known $q$-plane wave solutions. The present work thus extends the range of possible processes yielded by the NRT dynamics that admit an analytical, exact treatment. In the $q \to 1$ limit the $q$-Gaussian solutions correspond to the Gaussian wave packet solutions to the free particle linear Schrödinger equation. In the present work we also show that there are other families of nonlinear Schrödinger-like equations, besides the NRT one, exhibiting a dynamics compatible with the de Broglie relations. Remarkably, however, the existence of time dependent Gaussian-like wave packet solutions is a unique feature of the NRT equation not shared by the aforementioned, more general, families of nonlinear evolution equations.
We determine conditions under which a generic gauge invariant nonautonomous and inhomogeneous nonlinear partial differential equation in the two-dimensional space-time continuum can be transform into standard autonomous forms. In addition to the nonlinear Schroedinger equation, important examples include the derivative nonlinear Schroedinger equation, the quintic complex Ginzburg-Landau equation, and the Gerdjikov-Ivanov equation. This approach provides a mathematical description of nonstationary media supporting unidimensional signal propagation and/or total field trapping. In particular, we …
We investigate the implication of the non-linear and non-local multi-particle Schroedinger-Newton equation for the motion of the mass centre of an extended multi-particle object, giving self-contained and comprehensible derivations. In particular, we discuss two opposite limiting cases. In the first case, the width of the centre-of-mass wave packet is assumed much larger than the actual extent of the object, in the second case it is assumed much smaller. Both cases result in non-linear deviations from ordinary free Schroedinger evolution for the centre of mass. On a general conceptual level we…
We study discrete Schroedinger operators with analytic potentials. In particular, we are interested in the connection between the absolutely continuous spectrum in the almost periodic case and the spectra in the periodic case. We prove a weak form of a precise conjecture relating the two. We also bound the measure of the spectrum in the periodic case in terms of the Lyapunov exponent in the almost periodic case. In the proofs, we use a partial generalization of Chambers' formula. As an additional application of this generalization, we provide a new proof of Herman's lower bound for the Lyapuno…
The asymptotic iteration method is used to find exact and approximate solutions of Schroedinger's equation for a number of one-dimensional trigonometric potentials (sine-squared, double-cosine, tangent-squared, and complex cotangent). Analytic and approximate solutions are obtained by first using a coordinate transformation to reduce the Schroedinger equation to a second-order differential equation with an appropriate form. The asymptotic iteration method is also employed indirectly to obtain the terms in perturbation expansions, both for the energies and for the corresponding eigenfunctions.
In the quantum frame, for 3-dimensional space, in the two body problem case, we approach the Schrödinger equation (SE) taking in account the potential: Vq(r)=Dr^2+(A/r)+(B/r^2) called by us quasi-harmonic potential with the centrifugal type term, with D,A,B >0 and D<<1. We use Laplace transform method (LTM) and we find for the first time an analytic solution of the Vq-potential problem. Namely, using directly and inverse Laplace transformations, we obtain the complete forms of the energy eigenvalues and wave functions. Furthermore, for this potential Vq, we make considerations about critical o…
We consider two-dimensional Schroedinger equations with honeycomb potentials and slow time-periodic forcing of the form: $$iψ_t (t,x) = H^\varepsilon(t)ψ=\left(H^0+2i\varepsilon A (\varepsilon t) \cdot \nabla \right)ψ,\quad H^0=-Δ+V (x) .$$ The unforced Hamiltonian, $H^0$, is known to generically have Dirac (conical) points in its band spectrum. The evolution under $H^\varepsilon(t)$ of {\it band limited Dirac wave-packets} (spectrally localized near the Dirac point) is well-approximated on large time scales ($t\lesssim \varepsilon^{-2+}$) by an effective time-periodic Dirac equation with a ga…
We analytically study rogue-wave (RW) solutions and rational solitons of an integrable fifth-order nonlinear Schrödinger (FONLS) equation with three free parameters. It includes, as particular cases, the usual NLS, Hirota, and Lakshmanan-Porsezian-Daniel (LPD) equations. We present continuous-wave (CW) solutions and conditions for their modulation instability in the framework of this model. Applying the Darboux transformation to the CW input, novel first- and second-order RW solutions of the FONLS equation are analytically found. In particular, trajectories of motion of peaks and depressions o…
This article is concerned with the linearisation around a dark soliton solution of the nonlinear Schrödinger equation. Crucially, we present analytic expressions for the four linearly-independent zero eigenvalue solutions (also known as Goldstone modes) to the linearised problem. These solutions are then used to construct a Greens matrix which gives the first-order spatial response due to some perturbation. Finally we apply this Greens matrix to find the correction to the dark-soliton wavefunction of a Bose-Einstein condensate in the presence of fluctuations.
The Nonlinear Schroedinger Equation (NLSE) with a random potential is motivated by experiments in optics and in atom optics and is a paradigm for the competition between the randomness and nonlinearity. The analysis of the NLSE with a random (Anderson like) potential has been done at various levels of control: numerical, analytical and rigorous. Yet, this model equation presents us with a highly inconclusive and often contradictory picture. We will describe the main recent results obtained in this field and propose a list of specific problems to focus on, that we hope will enable to resolve th…
We represent low dimensional quantum mechanical Hamiltonians by moderately sized finite matrices that reproduce the lowest O(10) boundstate energies and wave functions to machine precision. The method extends also to Hamiltonians that are neither Hermitian nor PT symmetric and thus allows to investigate whether or not the spectra in such cases are still real. Furthermore, the approach is especially useful for problems in which a position-dependent mass is adopted, for example in effective-mass models in solid-state physics or in the approximate treatment of coupled nuclear motion in molecular …
Consider the H^{1/2}-critical Schroedinger equation with a cubic nonlinearity in R^3, i \partial_t ψ+ Δψ+ |ψ|^2 ψ= 0. It admits an eight-dimensional manifold of periodic solutions called solitons e^{i(Γ+ vx - t|v|^2 + α^2 t)} ϕ(x-2tv-D, α), where ϕ(x, α) is a positive ground state solution of the semilinear elliptic equation -Δϕ+ α^2ϕ= ϕ^3. We prove that in the neighborhood of the soliton manifold there exists a H^{1/2} real analytic manifold N of asymptotically stable solutions of the Schroedinger equation, meaning they are the sum of a moving soliton and a dispersive term. Furthermore, a sol…
We show that a wide class of quantum systems with translational invariance can host dispersionless, soliton-like, wave packets. We focus on the setting where the effective, two-dimensional Hamiltonian acquires the form of the Dirac operator. The proposed framework for construction of the dispersionless wave packets is illustrated on silicene-like systems with topologically nontrivial effective mass. Our analytical predictions are accompanied by a numerical analysis and possible experimental realizations are discussed.
