Conditional and Lie symmetries of semi-linear 1D Schrödinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrödinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra (conf3)C. We consider non-hermitian re...

The realizations of the Lie algebra corresponding to the dynamical symmetry group SO(2, 1) of the Schrödinger equations for the Morse and the V = u2 +1/u2 potentials were known to be related by a canonical transformation. q–deformed analog of this transformation connecting two different realizations of the slq(2) algebra is presented. By the virtue of the q–canonical transformation a q–deformed...

We study the Strang splitting scheme for quasilinear Schrödinger equations. We establish the convergence of the scheme for solutions with small initial data. We analyze the linear instability of the numerical scheme, which explains the numerical blow-up of large data solutions and connects to analytical breakdown of regularity of solutions to quasilinear Schrödinger equations. Numerical tests a...

We consider one-dimensional difference Schrödinger equations

where u = u(t, x) : R → C is a complex-valued wave function, both λ 6= 0 and k > 5 are real numbers. A great deal of interesting research has been devoted to the mathematical analysis for the derivative nonlinear Schrödinger equations [3, 4, 6, 7, 8, 9, 10, 11, 13, 18, 21]. In [13], C. E. Kenig, G. Ponce and L. Vega studied the local existence theory for the Cauchy problem of the derivative non...

We consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or, at least, of the associated physical observables, is a formidable ma...

We give a systematic method for discretising Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure, also preserves the correct monotonic decrease of energy. The method is illustrated by many examples. In the Hamiltonian case these include: the sine-Gordon, ...

Using WKB methods for very small times, we prove some instability phenomena for semi-classical (linear or) nonlinear Schrödinger equations. The main step of the analysis consists in reducing the problem to an ordinary differential equation. The solution to this o.d.e. is explicit, and the instability mechanism is due to the presence of the semi-classical parameter. For nonlin-ear equations, our...

We study the scattering theory for the nonlinear Schrödinger equations with cubic and quadratic nonlinearities in one and two space dimensions, respectively. For example, the nonlinearities are sum of gauge invariant term and non-gauge invariant terms such as λ0|u|2u + λ1u + λ2uū + λ3ū in one dimensional case, where λ0 ∈ R and λ1, λ2, λ3 ∈ C. The scattering theory for these equations belongs to...

In this paper we present a WKB approximation for sphericallysymmetric solutions of the Schrödinger-Newton equations. These are nonlinear modifications of the ordinary Schrödinger equation involving gravitational selfinteraction of the wavefunction. Applying the WKB procedure leads to two different nonlinear differential equations for the gravitational potential U for positive and negative value...