Standing waves of the quintic NLS equation on the tadpole graph
- PDF / 635,137 Bytes
- 31 Pages / 439.37 x 666.142 pts Page_size
- 12 Downloads / 162 Views
Calculus of Variations
Standing waves of the quintic NLS equation on the tadpole graph Diego Noja1 · Dmitry E. Pelinovsky2 Received: 6 January 2020 / Accepted: 31 July 2020 © The Author(s) 2020
Abstract The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schrödinger equation with quintic power nonlinearity equipped with the Neumann–Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency ω ∈ (−∞, 0) is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in L 6 . The set of standing waves includes the set of ground states, which are the global minimizers of the energy at constant mass (L 2 -norm), but it is actually wider. While ground states exist only for a certain interval of masses, the standing waves exist for every ω ∈ (−∞, 0) and correspond to a bigger interval of masses. It is proven that there exist critical frequencies ω1 and ω0 with −∞ < ω1 < ω0 < 0 such that the standing waves are the ground state for ω ∈ [ω0 , 0), local constrained minima of the energy for ω ∈ (ω1 , ω0 ) and saddle points of the energy at constant mass for ω ∈ (−∞, ω1 ). Proofs make use of the variational methods and the analytical theory for differential equations. Mathematics Subject Classification 35Q55 · 81Q35 · 35R02
1 Introduction The analysis of nonlinear PDEs on metric graphs has recently attracted a certain attention [30]. One of the reason is potential applicability of this analysis to physical models such as Bose-Einstein condensates trapped in narrow potentials with T-junctions or X-junctions, or networks of optical fibers. Another reason is the possibility to rigorously prove a complicated
D. Noja has received funding for this project from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant No 778010 IPaDEGAN. D.E. Pelinovsky acknowledges the support of the NSERC Discovery grant.
B
Diego Noja [email protected] Dmitry E. Pelinovsky [email protected]
1
Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via R. Cozzi 55, 20126 Milano, Italy
2
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada 0123456789().: V,-vol
123
173
Page 2 of 31
D. Noja, D. Pelinovsky
behavior of the standing waves due to the interplay between geometry and nonlinearity, which is hardly accessible in higher dimensional problems. The most studied nonlinear PDE on a metric graph G is the nonlinear Schrödinger (NLS) equation with power nonlinearity, which we take in the following form: i
d Ψ = ΔΨ + ( p + 1)|Ψ |2 p Ψ , dt
(1.1)
where the wave function Ψ (t, ·) is defined componentwise on edges of the graph G subject to suitable boundary conditions at vertices of the graph G . The Laplace operator Δ and the power nonlinearity are also defined componentwise. The natural Neumann–Kirchhoff boundary conditions are typically added at the vertices to ensure that Δ is self
Data Loading...
