Exact solutions toD-dimensional Schrödinger equation with a pseudoharmonic oscillator

The exact solutions to the Schrödinger equation with a pseudoharmonic oscillator in an arbitrary dimensionD is presented.     

Exact spatial soliton solution for nonlinear Schrödinger equation with a type of transverse nonperiodic modulation

In this paper, we analyze (2 + 1)D nonlinear Schrödinger (NLS) equation based on a type of nonperiodic modulation of linear refractive index in the transverse direction. We obtain an exact solution in explicit form for the (2 + 1)D nonlinear Schrödinger (NLS) equation with the nonperiodic modulation. Finally, the stability of the solution is discussed numerically, and the results reveal that the solution is stable to the finite initial perturbations.     

On the fluid approximation to a nonlinear Schrödinger equation

We present a heuristic proof that the nonlinear Schrödinger equation (NLS) - in 2 + 1 dimensions has a family of solutions which can be well approximated by a collection of point vortices for a planar incompressible fluid. The novelty of our approach is that we begin with a representation of the NLS as a compressible perturbation of Euler's equations for hydrodynamics.     

非球谐振子势Schr(o)dinger方程的精确解

将非球谐振子势V(r)=ar2+br4+cr6径向波函数展开为指数函数与多项式函数的乘积,应用多项式函数的系数关系确定了体系的能级和波函数.结果表明,体系处于束缚态时,势参数a,b,c必须满足一定的约束条件.     

PT symmetry in a fractional Schrödinger equation

We investigate the fractional Schrödinger equation with a periodic ‐symmetric potential. In the inverse space, the problem transfers into a first‐order nonlocal frequency‐delay partial differential equation. We show that at a critical point, the band structure becomes linear and symmetric in the one‐dimensional case, which results in a nondiffracting propagation and conical diffraction of input beams. If only one channel in the periodic potential is excited, adjacent channels become uniformly excited along the propagation direction, which can be used to generate laser beams of high power and narrow width. In the two‐dimensional case, there appears conical diffraction that depends on the competition between the fractional Laplacian operator and the ‐symmetric potential. This investigation may find applications in novel on‐chip optical devices.

     

Spectral splitting method for nonlinear Schrödinger equations with singular potential

We consider the time-dependent one-dimensional nonlinear Schrödinger equation with pointwise singular potential. By means of spectral splitting methods we prove that the evolution operator is approximated by the Lie evolution operator, where the kernel of the Lie evolution operator is explicitly written. This result yields a numerical procedure which is much less computationally expensive than multi-grid methods previously used. Furthermore, we apply the Lie approximation in order to make some numerical experiments concerning the splitting of a soliton, interaction among solitons and blow-up phenomenon.     

Model of a relativistic oscillator in a generalized Schrödinger picture

In a generalized Schrödinger picture, we consider the motion of a relativistic particle in an external field (like in the case of a harmonic oscillator). In this picture the analogs of the Schrödinger operators of the particle are independent of both the time and the space coordinates. These operators induce operators which are related to Killing vectors of the Anti de Sitter (AdS) space. We also consider the nonrelativistic limit.     

Exact solutions of a nonpolynomially nonlinear Schrödinger equation

A nonlinear generalisation of Schrödinger's equation had previously been obtained using information-theoretic arguments. The nonlinearities in that equation were of a nonpolynomial form, equivalent to the occurrence of higher-derivative nonlinear terms at all orders. Here we construct some exact solutions to that equation in 1+1$1+1$ dimensions. On the half-line, the solutions resemble (exponentially damped) Bloch waves even though no external periodic potential is included. The solutions are nonperturbative as they do not reduce to solutions of the linear theory in the limit that the nonlinearity parameter vanishes. An intriguing feature of the solutions is their infinite degeneracy: for a given energy, there exists a very large arbitrariness in the normalisable wavefunctions. We also consider solutions to a q-deformed version of the nonlinear equation and discuss a natural discretisation implied by the nonpolynomiality. Finally, we contrast the properties of our solutions with other solutions of nonlinear Schrödinger equations in the literature and suggest some possible applications of our results in the domains of low-energy and high-energy physics.     

Exact and explicit solutions to the discrete nonlinear Schrödinger equation with a saturable nonlinearity

We analyze the discrete nonlinear Schrödinger equation with a saturable nonlinearity through the (G^′/G)-expansion method to present some improved results. Three types of analytic solutions with arbitrary parameters are constructed; hyperbolic, trigonometric, and rational which have not been explicitly computed before.     

Solitary wave solutions as a signature of the instability in the discrete nonlinear Schrödinger equation

The effect of instability on the propagation of solitary waves along one-dimensional discrete nonlinear Schrödinger equation with cubic nonlinearity is revisited. A self-contained quasicontinuum approximation is developed to derive closed-form expressions for small-amplitude solitary waves. The notion that the existence of nonlinear solitary waves in discrete systems is a signature of the modulation instability is used. With the help of this notion we conjecture that instability effects on moving solitons can be qualitative estimated from the analytical solutions. Results from numerical simulations are presented to support this conjecture.     

Exact solutions of the Kemmer equation for a Dirac oscillator

Exact solutions of Kemmer equation for charged, massive, spin-1 particles in the Dirac oscillator potential have been found. The eigensolutions of this potential have been calculated and discussed in both natural and unnatural parities.     

Asymmetric bound states via the quadrupled Schrödinger equation

An efficient iterative method of construction of bound states in an asymmetric potential well is suggested and tested on V(x) = ax² + bx³ + cx⁴.     

Exact solutions to three-dimensional time-dependent Schrödinger equation

With a view to obtain exact analytic solutions to the time-dependent Schrödinger equation for a few potentials of physical interest in three dimensions, transformation-group method is used. Interestingly, the integrals of motion in the new coordinates turn out to be the desired invariants of the systems.     

质量和频率随时间变化的谐振子的经典和量子精确解

对质量和频率含时的谐振子用新的简单方法分别求出了其经典运动方程和量子薛定谔方程的精确解。     

Exact X-wave solutions of the hyperbolic nonlinear Schrödinger equation with a supporting potential

We find exact solutions of the two- and three-dimensional nonlinear Schrödinger equation with a supporting potential. We focus in the case where the diffraction operator is of the hyperbolic type and both the potential and the solution have the form of an X-wave. Following similar arguments, several additional families of exact solutions can also can be found irrespectively of the type of the diffraction operator (hyperbolic or elliptic) or the dimensionality of the problem. In particular we present two such examples: The one-dimensional nonlinear Schrödinger equation with a stationary and a “breathing” potential and the two-dimensional nonlinear Schrödinger with a Bessel potential.     

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves. This is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing-wave solution is stable in and unstable in under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the nonradial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.     

Exact analytical solutions for the generalized non-integrable nonlinear Schrödinger equation with varying coefficients

Analytical solutions are reported for the generalized non-integrable nonlinear Schrödinger equation with varying coefficients using the similarity transformation and tri-function method, which involve three free functions of spaces to generate abundant wave structures. Three types of free functions are chosen to exhibit the corresponding nonlinear wave propagations.