Failure of scattering to standing waves for a Schrödinger equation with long-range nonlinearity on star graph

Failure of scattering for the NLSE with a point interaction in dimension two and three

“…Among the models included, the authors stress the case of the NLS equation with a delta potential in one dimension, a quite well studied model in the last decades, with many results ranging from well posedness, to scattering and orbital and asymptotic stability of standing waves (see the bibliography in [19]). Further extensions of the result for the NLS equation with a delta potential have also been given in the (still one dimensional) case of the NLS equation on star graphs [5,6]. In particular it is implicit in [5] the case of a line with a general point interaction, not restricted to delta potentials.…”

“…The nonlinearities with this property are called long range nonlinearities, because in a sense they mimic the effect of a long range potential in the linear equation. They are given by the condition 1 < p ⩽ 1 + 2 n , corresponding to 1 < p ⩽ 2 for n = 2 and 1 < p ⩽ 5 3 for n = 3. Absence of scattering in this framework has been well known since a long time, starting with the original papers [7,15,23]; see also [10], theorem 7.5.2.…”

Asymptotic behavior for the long-range nonlinear Schrödinger equation on star graph with the Kirchhoff boundary condition

“…To derive the factorization formula, we apply the Fourier transform F with respect to ∆ K derived by Weder [16], which is an extension of the usual Fourier transform on the line R. We also have an interest in the asymptotic behavior of the solutions in the case of 1 ≤ p ≤ 2. We will show that the scattering to the free solution fails when 1 ≤ p ≤ 2 by assuming more regularity than in the previous paper [3]. The proof is based on [5].…”

“…Since the star graph is the connected half-lines (the star graph with 2-edges is just a line), the critical exponent is expected to be p = 2 as the case of the line R. Yoshinaga [18] proved that the solutions of (1.3) scatter to the free solution when p > 2, whose argument is based on [17]. (See also [3,Remark 3] for the precise statement.) Recently, the first, second, and fourth authors [3] proved the failure of scattering when 0 < p < 1 (In fact, they treated the nonlinear Schrödinger equation with more general boundary conditions).…”

“…(See also [3,Remark 3] for the precise statement.) Recently, the first, second, and fourth authors [3] proved the failure of scattering when 0 < p < 1 (In fact, they treated the nonlinear Schrödinger equation with more general boundary conditions).…”