We construct 2-solitary wave solutions with logarithmic distance to the nonlinear Schrödinger equation,
In the integrable case ( and ), the existence of such solutions is known by inverse scattering (E. Olmedilla, Multiple pole solutions of the nonlinear Schrödinger equation, Physica D 25 (1987) 330–346; T. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34 (1972) 62–69). The mass-critical case exhibits a specific behavior related to blow-up, previously studied in Y. Martel, P. Raphaël (Strongly interacting blow up bubbles for the mass critical NLS, Ann. Sci. Éc. Norm. Supér. 51 (2018) 701–737).
On construit des solutions au problème de la propagation de deux ondes solitaires avec distance logarithmique de Schrödinger non linéaire,
Dans le cas intégrable ( et ), l'existence d'une telle solution est connue par la méthode dite d'inverse scaterring (E. Olmedilla, Multiple pole solutions of the nonlinear Schrödinger equation, Physica D 25 (1987) 330–346 ; T. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34 (1972) 62–69). Le cas d'une masse critique introduit un comportement spécifique lié à l'explosion, qui a été étudié précédemment par Y. Martel et P. Raphaël (Strongly interacting blow up bubbles for the mass critical NLS, Ann. Sci. Éc. Norm. Supér. 51 (2018) 701–737).
Accepted:
Published online:
Tiến Vinh Nguyễn 1
@article{CRMATH_2019__357_1_13_0, author = {Tiến Vinh Nguyễn}, title = {Existence of multi-solitary waves with logarithmic relative distances for the {NLS} equation}, journal = {Comptes Rendus. Math\'ematique}, pages = {13--58}, publisher = {Elsevier}, volume = {357}, number = {1}, year = {2019}, doi = {10.1016/j.crma.2018.11.012}, language = {en}, }
Tiến Vinh Nguyễn. Existence of multi-solitary waves with logarithmic relative distances for the NLS equation. Comptes Rendus. Mathématique, Volume 357 (2019) no. 1, pp. 13-58. doi : 10.1016/j.crma.2018.11.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.11.012/
[1] Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schrödinger Operators, Mathematical Notes, vol. 29, Princeton University Press, Princeton, NJ, USA, 1982
[2] Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, New York University, New York, 2003
[3] The Cauchy problem for the critical nonlinear Schrödinger equation in , Nonlinear Anal., Volume 14 (1990) no. 10, pp. 807-836
[4] Construction of multi-soliton solutions for the -supercritical gKdV and NLS equations, Rev. Mat. Iberoam., Volume 27 (2011) no. 1, pp. 273-302
[5] Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension, Discrete Contin. Dyn. Syst., Volume 34 (2014), pp. 1961-1993
[6] Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal., Volume 18 (2009) no. 6, pp. 1787-1840
[7] Threshold solutions for the focusing 3D cubic Schrödinger equation, Rev. Mat. Iberoam., Volume 26 (2010), pp. 1-56
[8] Equation of motion for interacting pulses, Phys. Rev. E, Volume 50 (1994), pp. 4672-4678
[9] Hamiltonian Methods in the Theory of Solitons, Springer-Verlag, 2007
[10] On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., Volume 32 (1979), pp. 1-32
[11] Interactions of solitons in non-integrable systems: direct perturbation method and applications, Physica D, Volume 3 (1981) no. 1–2, pp. 428-438
[12] Stability theory of solitary waves in the presence of symmetry, J. Funct. Anal., Volume 197 (1987), pp. 74-160
[13] Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system, Commun. Pure Appl. Math., Volume 43 (1990), pp. 299-333
[14] Flat blow-up in one-dimensional semilinear heat equations, Differ. Integral Equ., Volume 5 (1992), pp. 973-997
[15] Construction of two-bubble solutions for the energy-critical NLS, Anal. PDE, Volume 10 (2017) no. 8, pp. 1923-1959
[16] A perturbational approach to the two-soliton system, Physica D, Volume 3 (1981) no. 1–2, pp. 487-502
[17] Two-soliton solutions to the three-dimensional gravitational Hartree equation, Commun. Pure Appl. Math., Volume 62 (2009) no. 11, pp. 1501-1550
[18] Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation, Nonlinearity, Volume 30 (2017) no. 12, p. 4614
[19] Multiple pole solutions of the nonlinear Schrödinger equation, Physica D, Volume 25 (1987), pp. 330-346
[20] Multi-solitary waves for nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 23 (2006), pp. 849-864
[21] Description of two soliton collison for the quartic gKdV equation, Ann. of Math. (2), Volume 174 (2011), pp. 757-857
[22] Inelastic interaction of nearly equal solitons for the quartic gKdV equation, Invent. Math., Volume 183 (2011) no. 3, pp. 563-648
[23] Stability in of the sum of K solitary waves for some nonlinear Schrödinger equations, Duke Math. J., Volume 133 (2006), pp. 405-466
[24] Strongly interacting blow up bubbles for the mass critical NLS, Ann. Sci. Éc. Norm. Supér., Volume 51 (2018), pp. 701-737
[25] Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity, Commun. Math. Phys., Volume 129 (1990) no. 2, pp. 223-240
[26] On universality of blow-up profile for critical nonlinear Schrödinger equation, Invent. Math., Volume 156 (2004) no. 3, pp. 565-672
[27] The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math. (2), Volume 161 (2005) no. 1, pp. 157-222
[28] Stability and Blow up for the Nonlinear Schrödinger Equation, Lecture Notes for the Clay Summer School on Evolution Equations, ETH, Zurich, Switzerland, 2008
[29] Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc., Volume 24 (2011) no. 2, pp. 471-546
[30] Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., Volume 16 (1985), pp. 472-491
[31] Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math., Volume 39 (1986), pp. 51-68
[32] Nonlinear Waves in Integrable and Non-integrable Systems, SIAM, Philadelphia, PA, 2010
[33] Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP, Volume 34 (1972), pp. 62-69
Cited by Sources:
Comments - Policy