arXiv subjects

Erwan Le Quiniou

Publications and source records attributed to Erwan Le Quiniou.

3 recordsLinked to original sources

arXiv2026

Orbital stability of black solitons for quasilinear Schrödinger equations with nonzero conditions at infinity

We investigate the orbital stability of black solitons for a broad class of quasilinear Schrödinger equations in one space dimension, with nonzero boundary conditions at infinity. Namely, our framework handles general defocusing semilinear nonlinearities and focusing or defocusing quasilinear nonlinearities. First, we establish sufficient conditions on the quasi-linear nonlinearities ensuring the existence of a local branch of finite-energy solitons parameterized by their speed. Within this branch, the black soliton, also called kink, corresponds to the stationary solution. Our main result is the orbital stability of the black soliton in the energy space, provided that the Vakhitov-Kolokolov (VK) slope condition holds; namely, that the derivative of the momentum with respect to the speed is negative at zero. Moreover, we derive an explicit formula for verifying this VK condition. The proof relies on the analysis of a carefully designed variational problem, which allows us to control the sup-norm of the evolution of a perturbation of the kink in terms of the energy and momentum, both of which are conserved by the flow. A delicate part of the argument is the analysis of minimizing sequences for this variational problem, since the infimum is not attained.

math.AP↗

arXiv2025

Stability and instability of the quasilinear Gross--Pitaevskii dark solitons

We study a quasilinear Schrödinger equation with nonzero conditions at infinity. In previous works, we obtained a continuous branch of traveling waves, given by dark solitons indexed by their speed. Neglecting the quasilinear term, one recovers the Gross--Pitaevskii equation, for which the branch of dark solitons is stable. Moreover, Z.~Lin showed that the Vakhitov--Kolokolov~(VK) stability criterion (in terms of the momentum of solitons) holds for general semilinear equations with nonvanishing conditions at infinity. In the quasilinear case, we prove that the VK stability criterion still applies, by generalizing Lin's arguments. Therefore, we deduce that the branch of dark solitons is stable for weak quasilinear interactions. For stronger quasilinear interactions, a cusp appears in the energy-momentum diagram, implying the stability of fast waves and the instability of slow waves.

math.AP↗

arXiv2025

Exotic traveling waves for a quasilinear Schrödinger equation with nonzero background

We study a defocusing quasilinear Schrödinger equation with nonzero conditions at infinity in dimension one. This quasilinear model corresponds to a weakly nonlocal approximation of the nonlocal Gross--Pitaevskii equation, and can also be derived by considering the effects of surface tension in superfluids. When the quasilinear term is neglected, the resulting equation is the classical Gross-Pitaevskii equation, which possesses a well-known stable branch of subsonic traveling waves solution, given by dark solitons. Our goal is to investigate how the quasilinear term affects the traveling-waves solutions. We provide a complete classification of finite energy traveling waves of the equation, in terms of the two parameters: the speed and the strength of the quasilinear term. This classification leads to the existence of dark and antidark solitons, as well as more exotic localized solutions like dark cuspons, compactons, and composite waves, even for supersonic speeds. Depending on the parameters, these types of solutions can coexist, showing that finite energy solutions are not unique. Furthermore, we prove that some of these dark solitons can be obtained as minimizers of the energy, at fixed momentum, and that they are orbitally stable.

math.AP↗