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Register a new userAsymptotic stability of solitary waves for the $1d$ NLS with an attractive delta potential
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We consider the one-dimensional nonlinear Schrodinger equation with an attractive delta potential and mass-supercritical nonlinearity. This equation admits a one-parameter family of solitary wave solutions in both the focusing and defocusing cases. We establish asymptotic stability for all solitary waves satisfying a suitable spectral condition, namely, that the linearized operator around the solitary wave has a two-dimensional generalized kernel and no other eigenvalues or resonances. In particular, we extend our previous result beyond the regime of small solitary waves and extend the results of Fukuizumi-Ohta-Ozawa and Kaminaga-Ohta from orbital to asymptotic stability for a suitable family of solitary waves.
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We consider a Nonlinear Schrodinger Equation with a very general non linear term and with a trapping $delta $ potential on the line. We then discuss the asymptotic behavior of all its small solutions, generalizing a recent result by Masaki et al. We give also a result of dispersion in the case of defocusing equations with a non--trapping delta potential.
We consider dispersion generalized nonlinear Schrodinger equations (NLS) of the form $i partial_t u = P(D) u - |u|^{2 sigma} u$, where $P(D)$ denotes a (pseudo)-differential operator of arbitrary order. As a main result, we prove symmetry results for traveling solitary waves in the case of powers $sigma in mathbb{N}$. The arguments are based on Steiner type rearrangements in Fourier space. Our results apply to a broad class of NLS-type equations such as fourth-order (biharmonic) NLS, fractional NLS, square-root Klein-Gordon and half-wave equations.
We prove the existence of a 2-parameter family of small quasi-periodic in time solutions of discrete nonlinear Schrodinger equation (DNLS). We further show that all small solutions of DNLS decouples to this quasi-periodic solution and dispersive wave.
The Degasperis-Procesi equation is an approximating model of shallow-water wave propagating mainly in one direction to the Euler equations. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations with the same asymptotic accuracy, and is integrable with the bi-Hamiltonian structure. In the present study, we establish existence and spectral stability results of localized smooth solitons to the Degasperis-Procesi equation on the real line. The stability proof relies essentially on refined spectral analysis of the linear operator corresponding to the second-order variational derivative of the Hamiltonian of the Degasperis-Procesi equation.
The Degasperis-Procesi equation is the integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the Desgasperis-Procesi (DP) equation on the real line. %extending our previous work on their spectral stability cite{LLW}. The main difficulty stems from the fact that the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the $L^2$-norm, which by itself can not bound the higher-order nonlinear terms in the Lagrangian. The remedy is to observe that, given a sufficiently smooth initial condition satisfying a measurable constraint, the $L^infty$ orbital norm of the perturbation is bounded above by a function of its $L^2$ orbital norm, yielding the orbital stability in the $L^2cap L^infty$ space.
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