Eduard Kirr

Symmetry-Breaking Bifurcation in the Nonlinear Schrödinger Equation with Symmetric Potentials

We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter.

SYMMETRY-BREAKING BIFURCATION IN NONLINEAR SCHRODINGER/GROSS-PITAEVSKII EQUATIONS

We consider a class of nonlinear Schrodinger/Gross–Pitaeveskii (NLS-GP) equations, i.e., NLS with a linear potential. NLS-GP plays an important role in the mathematical modeling of nonlinear optical as well as macroscopic quantum phenomena (BEC). We obtain conditions for a symmetry-breaking bifurcation in a symmetric family of states as

On the Asymptotic Stability of Bound States in 2D Cubic Schrödinger Equation

{\cal N}

Parametrically Excited Hamiltonian Partial Differential Equations

, the squared

Asymptotic stability of ground states in 2D nonlinear Schrödinger equation including subcritical cases

L^2

Compressible two-phase flows by central and upwind schemes

norm (particle number, optical power), is increased. The bifurcating asymmetric state is a “mixed mode” which, near the bifurcation point, is approximately a superposition of symmetric and antisymmetric modes. In the special case where the linear potential is a double well with well-separation L, we estimate

Parametric resonance of ground states in the nonlinear Schrödinger equation

{\cal N}_{cr}(L)

Metastable States in Parametrically Excited Multimode Hamiltonian Systems

, the symmetry breaking threshold. Along the “lowest energy” symmetric branch, there is an exchange of stability from the symmetric to the asymmetric branch as

Diffusion of Power in Randomly Perturbed Hamiltonian Partial Differential Equations

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Dynamic Statistical Scaling in the Landau-de Gennes Theory of Nematic Liquid Crystals

is increased beyond