Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T13:24:08.134Z Has data issue: false hasContentIssue false
  • English
  • Français

Concentrating standing waves for Davey–Stewartson systems

Part of: Incompressible inviscid fluids Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  01 October 2021

Yi He  and
Xiao Luo
Yi He
Affiliation:
School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, P. R. China ([email protected])
Xiao Luo
Affiliation:
School of Mathematics, Hefei University of Technology, Hefei 230009, P. R. China ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey–Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey–Stewartson systems. We emphasize that with respect to the classical Schrödinger equation, the presence of a singular integral operator in the Davey–Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.

Keywords

ConcentrationDavey–Stewartson systemmulti-peak solutionssingular integral operator

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35B25: Singular perturbations 35J60: Nonlinear elliptic equations 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 6 , December 2022 , pp. 1411 - 1450
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ambrosetti, A., Badiale, M. and Cingolani, S.. Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 140 (1997), 285–300.CrossRefGoogle Scholar
Berestycki, H., Gallouët, T. and Kavian, O.. Équations de champs scalaires Euclidens non linéaires dans le plan. C.R. Acad. Sci. Paris Sér. I Math. 297 (1983), 307–310.Google Scholar
Berestycki, H. and Lions, P. L.. Nonlinear scalar field equations, I. Existence of a ground state. Arch. Ration. Mech. Anal. 82 (1983), 313–345.CrossRefGoogle Scholar
Berestycki, H. and Lions, P. L.. Nonlinear scalar field equations, II. Existence of infinitely many solutions. Arch. Ration. Mech. Anal. 82 (1983), 347–375.CrossRefGoogle Scholar
Brézis, H. and Lieb, E. H.. A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88 (1983), 486–490.CrossRefGoogle Scholar
Byeon, J. and Jeanjean, L.. Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. 185 (2007), 185–200.CrossRefGoogle Scholar
Byeon, J. and Jeanjean, L.. Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity. Discrete Contin. Dyn. Syst. Ser. A 19 (2007), 255–269.CrossRefGoogle Scholar
Byeon, J., Jeanjean, L. and Tanaka, K.. Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases. Commun. Partial Differ. Equ. 33 (2008), 1113–1136.CrossRefGoogle Scholar
Byeon, J. and Tanaka, K.. Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential. J. Eur. Math. Soc. 15 (2013), 1859–1899.CrossRefGoogle Scholar
Byeon, J. and Tanaka, K.. Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations. Mem. Amer. Math. Soc. 229 (2014), No. 1076.Google Scholar
Byeon, J. and Wang, Z.-Q.. Standing waves with a critical frequency for nonlinear Schrödinger equations II. Calc. Var. Partial Differ. Equ. 18 (2003), 207–219.CrossRefGoogle Scholar
Carles, R., Dumas, E. and Sparber, C.. Geometric optics and instability for NLS and Davey–Stewartson models. J. Eur. Math. Soc. 14 (2012), 1885–1921.CrossRefGoogle Scholar
Cingolani, S., Jeanjean, L. and Tanaka, K.. Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well. Calc. Var. Partial Differ. Equ. 53 (2015), 413–439.CrossRefGoogle Scholar
Cingolani, S. and Lazzo, N.. Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations. Topol. Methods Nonlinear Anal. 10 (1997), 1–13.CrossRefGoogle Scholar
Cipolatti, R.. On the existence of standing waves for a Davey–Stewartson system. Commun. Partial Differ. Equ. 17 (1992), 967–988.CrossRefGoogle Scholar
Cipolatti, R.. On the instability of ground states for a Davey–Stewartson system. Ann. Inst. Henri Poincaré Phys. Théor. 58 (1993), 85–104.Google Scholar
Coti Zelati, V. and Rabinowitz, P.. Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Amer. Math. Soc. 4 (1991), 693–727.CrossRefGoogle Scholar
Craig, W., Schanz, U. and Sulem, C.. The modulational regime of three-dimensional water waves and the Davey–Stewartson system. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1999), 131–175.Google Scholar
D'Aprile, T. and Ruiz, D.. Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems. Math. Z. 268 (2011), 605–634.CrossRefGoogle Scholar
D'Avenia, P., Pomponio, A. and Ruiz, D.. Semiclassical states for the nonlinear Schrödinger equation on saddle points of the potential via variational methods. J. Funct. Anal. 262 (2012), 4600–4633.CrossRefGoogle Scholar
Davey, A. and Stewartson, K.. On three-dimensional packets of surface waves. Proc. Roy. Soc. Lond. Ser. A 338 (1974), 101–110.Google Scholar
del Pino, M. and Felmer, P. L.. Local mountain pass for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4 (1996), 121–137.CrossRefGoogle Scholar
del Pino, M. and Felmer, P. L.. Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non. Linéaire 15 (1998), 127–149.CrossRefGoogle Scholar
Figueiredo, G. M., Ikoma, N. and Santos, J. R. Jr. Existence and concentration result for the Kirchhoff type equations with general nonlinearities. Arch. Ration. Mech. Anal. 213 (2014), 931–979.CrossRefGoogle Scholar
Floer, A. and Weinstein, A.. Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69 (1986), 397–408.CrossRefGoogle Scholar
Gan, Z. H. and Zhang, J.. Sharp threshold of global existence and instability of standing wave for a Davey–Stewartson system. Commun. Math. Phys. 283 (2008), 93–125.CrossRefGoogle Scholar
Ghidaglia, J. M. and Saut, J. C.. On the initial value problem for the Davey–Stewartson systems. Nonlinearity 3 (1990), 475–506.CrossRefGoogle Scholar
Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order, 2nd edn, Grundlehren Math. Wiss., vol. 224 (Berlin: Springer, 1983).Google Scholar
Gui, C.. Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method. Commun. Partial Differ. Equ. 21 (1996), 787–820.CrossRefGoogle Scholar
Guo, B. L. and Wang, B. X.. The Cauchy problem for Davey–Stewartson systems. Commun Pure Appl. Math. 52 (1999), 1477–1490.3.0.CO;2-N>CrossRefGoogle Scholar
Jeanjean, L. and Tanaka, K.. A remark on least energy solutions in ${\mathbb {R}}^{N}$. Proc. Amer. Math. Soc. 131 (2002), 2399–2408.CrossRefGoogle Scholar
Lieb, E. H. and Loss, M.. Analysis, 2nd edn, Graduate Studies in Mathematics, vol. 14 (Providence, RI: American Mathematical Society, 2001).Google Scholar
Lions, P. L.. The concentration-compactness principle in the calculus of variations. The locally compact case, part 2. Ann. Inst. H. Poincaré Anal. Non. Linéaire 2 (1984), 223–283.CrossRefGoogle Scholar
Oh, Y. G.. Existence of semi-classical bound states of nonlinear Schrödinger equations with potential on the class $( V )_{a}$. Commun. Partial Differ. Equ. 13 (1988), 1499–1519.CrossRefGoogle Scholar
Oh, Y. G.. On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 131 (1990), 223–253.CrossRefGoogle Scholar
Ohta, M.. Blow-up solutions and strong instability of standing waves for the generalized Davey–Stewartson system in $\mathbb {R}^{2}$. Ann. Inst. Henri Poincaré Phys. Théor. 63 (1995), 111–117.Google Scholar
Ohta, M.. Instability of standing waves for the generalized Davey–Stewartson system. Ann. Inst. Henri Poincaré Phys. Théor. 62 (1995), 69–80.Google Scholar
Ozawa, T.. Exact blow-up solutions to the Cauchy problem for the Davey–Stewartson systems. Proc. Roy. Soc. Lond. Ser. A 436 (1992), 345–349.Google Scholar
Pucci, P. and Serrin, J.. A general variational identity. Indiana Univ. Math. J. 35 (1986), 681–703.CrossRefGoogle Scholar
Rabinowitz, P.. On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43 (1992), 270–291.CrossRefGoogle Scholar
Wang, X.. On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 153 (1993), 229–244.CrossRefGoogle Scholar
Weinstein, M.. Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87 (1983), 567–576.CrossRefGoogle Scholar