2,123 research outputs found
On linear instability of solitary waves for the nonlinear Dirac equation
We consider the nonlinear Dirac equation, also known as the Soler model:
i\p\sb t\psi=-i\alpha \cdot \nabla \psi+m \beta \psi-f(\psi\sp\ast \beta \psi)
\beta \psi, , , , f\in
C\sp 2(\R), where , , and are
Hermitian matrices which satisfy , , . We study the spectral stability of solitary wave solutions
. We study the point spectrum of linearizations at
solitary waves that bifurcate from NLS solitary waves in the limit , proving that if , then one positive and one negative eigenvalue are
present in the spectrum of the linearizations at these solitary waves with
sufficiently close to , so that these solitary waves are linearly
unstable. The approach is based on applying the Rayleigh--Schroedinger
perturbation theory to the nonrelativistic limit of the equation. The results
are in formal agreement with the Vakhitov--Kolokolov stability criterion.Comment: 17 pages. arXiv admin note: substantial text overlap with
arXiv:1203.3859 (an earlier 1D version
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
- …