Stability of standing waves for NLS-log equation with $$\varvec{\delta }$$-interaction

Abstract

Abstract. We study analytically the orbital stability of the standing waves with a peak-Gausson profile for a nonlinear logarithmic Schrö-dinger equation with δ-interaction (attractive and repulsive). A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing wave. This is overcome by the perturbation method, the continuation arguments, and the theory of extensions of symmetric operators.Mathematics Subject Classification (2010). Primary 35Q51, 35J61; Secondary 47E05.

On stability properties of the Cubic-Quintic Schródinger equation with <inline-formula><tex-math id="M1">\begin{document}$\delta$\end{document}</tex-math></inline-formula>-point interaction

“…From (3) follows L 2,Z (φ ω,Z ) = 0. Thus, since φ ω,Z > 0 we obtain from the Sturm-Liouville oscillation theory extended to operators with point interaction in Angulo&Goloshchapova [12,13] that zero is a simple isolated eigenvalue, the remains of the spectrum is contained in [δ, +∞) for δ > 0. Moreover, from Weyl's theorem (see Reed&Simon [48]) we obtain the affirmation on the essential spectrum.…”

“…For finding this number we will use perturbation theory and we will follow the ideas in Le Coz et.al [22]. We also give an alternative approach based in the extension theory for symmetric operators of von Neumman and Krein established recently by Angulo&Goloshchapova ( [12], [13]) for finding this index at least in the case Z > 0 (see Appendix below). Proof.…”

“…The case of Schrödinger models on star graphs with δ conditions on the vertex also have been studied recently in Adami et.al ( [1,2] and references therein) and An-gulo&Goloshchapova ( [10,11,12]). The case of either other defect type or nonlinearity have been studied in [3,9,12,13,43]. The specific case λ 1 = 2, λ 2 = −1 and Z > 0 was recently considered by Genoud&Malomed&Weishaupl in [30].…”

Solving forward and inverse problems of the logarithmic nonlinear Schrödinger equation with PT-symmetric harmonic potential via deep learning

“…Eq. ( 6) can be applied to quantum mechanics [51][52][53], dissipative systems [54], nuclear physics [55,56], quantum optics [57], magma transport phenomena [58], open quantum systems, effective quantum gravity, theory of superfluidity and Bose-Einstein condensation [59][60][61][62][63].…”

“…Eq. (6) can be applied to quantum mechanics [51][52][53], dissipative systems [54], nuclear physics [55,56], quantum optics [57], magma transport phenomena [58], open quantum systems, effective quantum gravity, theory of superfluidity and Bose-Einstein condensation [59][60][61][62][63]. We here use the above-mentioned PINNs deep learning method [11] to study the data-driven solutions of the dimensionless LNLS equation with the complex PT -symmetric potential…”