No Access

Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations

Yakine Bahri

Department of Mathematics and Statistics, University of Victoria, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada

Pacific Institute for Mathematical Sciences, 4176-2207 Main Mall, Vancouver, BC V6T 1Z4, Canada

E-mail Address: ybahri@uvic.ca

Search for more papers by this author

Slim Ibrahim

Department of Mathematics and Statistics, University of Victoria, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada

Pacific Institute for Mathematical Sciences, 4176-2207 Main Mall, Vancouver, BC V6T 1Z4, Canada

E-mail Address: ibrahims@uvic.ca

Corresponding author.

Search for more papers by this author

, and

Hiroaki Kikuchi

Department of Mathematics, Tsuda University, 2-1-1 Tsuda-machi, Kodaira-shi, Tokyo 187-8577, Japan

E-mail Address: hiroaki@tsuda.ac.jp

Search for more papers by this author

https://doi.org/10.1142/S0219199720500583Cited by:6 (Source: Crossref)

Abstract

In this paper, we study solitary wave solutions of the Cauchy problem for Half-wave-Schrödinger equation in the plane. First, we show the existence and the orbital stability of the ground states. Second, we prove that given any speed $v$ , traveling wave solutions exist and converge to the zero wave as the velocity tends to $1$ . Finally, we solve the Cauchy problem for initial data in $L_{x}^{2} H_{y}^{s} (ℝ^{2})$ , with $s > \frac{1}{2}$ . The critical case $s = \frac{1}{2}$ still stands as an interesting open problem.

Keywords:

AMSC: 35J20, 35B09, 35Q53, 35Q55

References

1. J. Bellazzini, V. Georgiev and N. Visciglia, Traveling waves for the quartic focusing Half-wave equation in one space dimension, preprint (2018); arXiv:1804.07075. Google Scholar
2. J. M. Bouclet and N. Tzvetkov , Strichartz estimates for long range perturbations, Amer. J. Math. 129 (2007) 1565–1609. Crossref, Web of Science, Google Scholar
3. H. Brezis and E. H. Lieb , A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486–490. Crossref, Web of Science, Google Scholar
4. D. Cai, A. J. Majda, D. W. McLaughlin and E. G. Tabak , Dispersive wave turbulence in one dimension, Phys. D 152/153 (2001) 551–572. Advances in nonlinear mathematics and science. Crossref, Web of Science, Google Scholar
5. T. Cazenave , Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, Vol. 10 (New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003). Crossref, Google Scholar
6. T. Cazenave and P.-L. Lions , Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982) 549–561. Crossref, Web of Science, Google Scholar
7. T. Cazenave and F. Weissler , The Cauchy problem for the nonlinear Schrödinger equation in $H^{1}$ , Manuscripta Math. 61 (1988) 477–494. Crossref, Web of Science, Google Scholar
8. B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, preprint (2011); arXiv:1104.1114. Google Scholar
9. A. Elgart and B. Schlein , Mean field dynamics of boson stars, Comm. Pure Appl. Math. 60 (2007) 500–545. Crossref, Web of Science, Google Scholar
10. A. Esfahani and A. Pastor , Sharp constant of an anisotropic Gagliardo–Nirenberg-type inequality and applications, Bull. Braz. Math. Soc. (N.S.) 48 (2017) 171–185. Crossref, Web of Science, Google Scholar
11. A. Esfahani, A. Pastor and J. Bona , Stability and decay properties of solitary-wave solutions to the generalized BO–ZK equation, Adv. Differential Equations 20 (2015) 801–834. Google Scholar
12. R. L. Frank and E. Lenzmann , Uniqueness of non-linear ground states for fractional Laplacians in $ℝ$ , Acta Math. 210 (2013) 261–318. Crossref, Web of Science, Google Scholar
13. R. L. Frank, E. Lenzmann and L. Silvestre , Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. 69 (2016) 1671–1726. Crossref, Web of Science, Google Scholar
14. J. Fröhlich and E. Lenzmann , Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math. 60 (2007) 1691–1705. Crossref, Web of Science, Google Scholar
15. P. Gérard, E. Lenzmann, O. Pocovnicu and P. Raphaël , A two-soliton with transient turbulent regime for the cubic Half-wave equation on the real line, Ann. PDE 4 (2018) 166 pp. Crossref, Google Scholar
16. J. Ginibre and G. Velo , The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985) 309–327. Crossref, Web of Science, Google Scholar
17. J. M. Gonçalves Ribeiro , Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field, Ann. Inst. H. Poincaré Phys. Théor. 54 (1991) 403–433. Web of Science, Google Scholar
18. L. Grafakos and S. Oh , The Kato–Ponce inequality, Comm. Partial Differential Equations 39 (2014) 1128–1157. Crossref, Web of Science, Google Scholar
19. M. Grillakis, J. Shatah and W. Strauss , Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987) 160–197. Crossref, Web of Science, Google Scholar
20. M. Keel and T. Tao , Endpoint Strichartz estimates, Amer. J. Math. 120 (1998) 955–980. Crossref, Web of Science, Google Scholar
21. K. Kirkpatrick, E. Lenzmann and G. Staffilani , On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys. 317 (2013) 563–591. Crossref, Web of Science, Google Scholar
22. H. Koch, D. Tataru and M. Visan , Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars, Vol. 45 (Springer, 2014). Crossref, Google Scholar
23. J. Krieger, E. Lenzmann and P. Raphaël , Nondispersive solutions to the $L^{2}$ -critical half-wave equation, Arch. Ration. Mech. Anal. 209 (2013) 61–129. Crossref, Web of Science, Google Scholar
24. P. I. Lizorkin , Multipliers of Fourier integrals in the spaces $L_{p, 𝜃}$ , Tr. Mat. Inst. Steklova 89 (1967) 231–248. Google Scholar
25. A. J. Majda, E. W. McLaughlin and E. G. Tabak , A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci. 7 (1997) 9–44. Crossref, Web of Science, Google Scholar
26. Y. J. Park , Fractional Rellich–Kondrachov compactness theorem, J. Chungcheong Math. Soc. 25 (2012) 527–529. Crossref, Google Scholar
27. D. Robert , Autour de l’approximation Semi-classique, Progress in Mathematics, Vol. 68 (Birkhäuser, Basel, 1987). Google Scholar
28. J. Shatah and W. Strauss , Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985) 173–190. Crossref, Web of Science, Google Scholar
29. N. Tzvetkov and N. Visciglia , Well-posedness and scattering for nonlinear Schrödinger equations on $ℝ^{d} \times 𝕋$ in the energy space, Rev. Mat. Iberoamericana 32 (2016) 1163–1188. Crossref, Web of Science, Google Scholar
30. H. Xu , Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation, Math. Z. 286 (2017) 443–489. Crossref, Web of Science, Google Scholar
31. K. Yajima , Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987) 415–426. Crossref, Web of Science, Google Scholar
32. M. Zworski , Semiclassical Analysis, Graduate Studies in Mathematics, Vol. 138 (American Mathematical Society, 2012). Crossref, Google Scholar