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Register a new userOn stability of standing waves of nonlinear Dirac equations
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Nabile Boussaid
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We consider the stability problem for standing waves of nonlinear Dirac models. Under a suitable definition of linear stability, and under some restriction on the spectrum, we prove at the same time orbital and asymptotic stability. We are not able to get the full result proved by Cuccagna for the nonlinear Schrodinger equation, because of the strong indefiniteness of the energy.
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We study the behavior of perturbations of small nonlinear Dirac standing waves. We assume that the linear Dirac operator of reference $H=D_m+V$ has only two double eigenvalues and that degeneracies are due to a symmetry of $H$ (theorem of Kramers). In this case, we can build a small 4-dimensional manifold of stationary solutions tangent to the first eigenspace of $H$. Then we assume that a resonance condition holds and we build a center manifold of real codimension 8 around each stationary solution. Inside this center manifold any $H^{s}$ perturbation of stationary solutions, with $s>2$, stabilizes towards a standing wave. We also build center-stable and center-unstable manifolds each one of real codimension 4. Inside each of these manifolds, we obtain stabilization towards the center manifold in one direction of time, while in the other, we have instability. Eventually, outside all these manifolds, we have instability in the two directions of time. For localized perturbations inside the center manifold, we obtain a nonlinear scattering result.
Continuing a line of investigation initiated in [11] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman-Schwinger type integral operators, we here examine the stability index, or sign of the first nonvanishing derivative at frequency zero of the characteristic determinant, an object that has found considerable use in the study by Evans function techniques of stability of standing and traveling wave solutions of partial differential equations (PDE) in one dimension. This leads us to the derivation of general perturbation expansions for analytically-varying modified Fredholm determinants of abstract operators. Our main conclusion, similarly in the analysis of the determinant itself, is that the derivative of the characteristic Fredholm determinant may be efficiently computed from first principles for integral operators with semi-separable integral kernels, which include in particular the general one-dimensional case, and for sums thereof, which latter possibility appears to offer applications in the multi-dimensional case. A second main result is to show that the multi-dimensional characteristic Fredholm determinant is the renormalized limit of a sequence of Evans functions defined in [23] on successive Galerkin subspaces, giving a natural extension of the one-dimensional results of [11] and answering a question of [27] whether this sequence might possibly converge (in general, no, but with renormalization, yes). Convergence is useful in practice for numerical error control and acceleration.
In this paper we deal with two dimensional cubic Dirac equations appearing as effective model in gapped honeycomb structures. We give a formal derivation starting from cubic Schrodinger equations and prove the existence of standing waves bifurcating from one band-edge of the linear spectrum.
We prove the existence of infinitely many non square-integrable stationary solutions for a family of massless Dirac equations in 2D. They appear as effective equations in two dimensional honeycomb structures. We give a direct existence proof thanks to a particular radial ansatz, which also allows to provide the exact asymptotic behavior of spinor components. Moreover, those solutions admit a variational characterization. We also indicate how the content of the present paper allows to extend our previous results for the massive case [5] to more general nonlinearities.
We study bifurcations and spectral stability of solitary waves in coupled nonlinear Schrodinger equations (CNLS) on the line. We assume that the coupled equations possess a solution of which one component is identically zero, and call it a $textit{fundamental solitary wave}$. By using a result of one of the authors and his collaborator, the bifurcations of the fundamental solitary wave are detected. We utilize the Hamiltonian-Krein index theory and Evans function technique to determine the spectral or orbital stability of the bifurcated solitary waves as well as as that of the fundamental one under some nondegenerate conditions which are easy to verify, compared with those of the previous results. We apply our theory to CNLS with a cubic nonlinearity and give numerical evidences for the theoretical results.
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