7,666 research outputs found
Second harmonic Hamiltonian: Algebraic and Schrödinger approaches
We study in detail the behavior of the energy spectrum for the second harmonic generation (SHG) and a family of corresponding quasi-exactly solvable Schrödinger potentials labeled by a real parameter b. The eigenvalues of this system are obtained by the polynomial deformation of the Lie algebra representation space. We have found the bi-confluent Heun equation (BHE) corresponding to this system in a differential realization approach, by making use of the symmetries. By means of a b-transformation from this second-order equation to a Schrödinger one, we have found a family of quasi-exactly solvable potentials. For each invariant n-dimensional subspace of the second harmonic generation, there are either n potentials, each with one known solution, or one potential with n-known solutions. Well-known potentials like a sextic oscillator or that of a quantum dot appear among them
Exact stationary solutions of the parametrically driven and damped nonlinear Dirac equation
Two exact stationary soliton solutions are found in the parametrically driven and damped nonlinear Dirac equation. The parametric force
considered is a complex ac force. The solutions appear when their frequencies are locked to half the frequency of the parametric force, and
their phases satisfy certain conditions depending on the force amplitude and on the damping coe cient. Explicit expressions for the charge,
the energy, and the momentum of these solutions are provided. Their stability is studied via a variational method using an ansatz with only
two collective coordinates. Numerical simulations con rm that one of the solutions is stable, while the other is an unstable saddle point.
Consequently, the stabilization of damped Dirac solitons can be achieved via time-periodic parametric excitations.Junta de AndalucÃa and Ministerio de EconomÃa y Competitividad of Spain FIS2017-89349-PMinisterio de Ciencia, Innovación y Universidades of Spain PGC2018-093998-BI0
Local controllability of 1D linear and nonlinear Schr\"odinger equations with bilinear control
We consider a linear Schr\"odinger equation, on a bounded interval, with
bilinear control, that represents a quantum particle in an electric field (the
control). We prove the controllability of this system, in any positive time,
locally around the ground state. Similar results were proved for particular
models (by the first author and with J.M. Coron), in non optimal spaces, in
long time and the proof relied on the Nash-Moser implicit function theorem in
order to deal with an a priori loss of regularity. In this article, the model
is more general, the spaces are optimal, there is no restriction on the time
and the proof relies on the classical inverse mapping theorem. A hidden
regularizing effect is emphasized, showing there is actually no loss of
regularity. Then, the same strategy is applied to nonlinear Schr\"odinger
equations and nonlinear wave equations, showing that the method works for a
wide range of bilinear control systems
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