Existence of multi-solitons for the focusing Logarithmic Non-Linear Schrödinger Equation

Ferriere, Guillaume

doi:10.1016/j.anihpc.2020.09.002

Ferriere, Guillaume ¹

¹ IMAG, Univ Montpellier, CNRS, Case courrier 051, Place Eugène Bataillon, 34090 Montpellier, France

Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 841-875.

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Résumé
Abstract

On considère l'équation de Schrödinger logarithmique (logNLS) en régime focalisant. Pour cette équation, les données initiales gaussiennes restent gaussiennes. En particulier, le Gausson - une fonction gaussienne indépendante du temps - est une solution orbitalement stable. Dans cet article, nous construisons des multi-solitons (ou multi-Gaussons) pour logNLS, avec estimées dans $H^{1} \cap F (H^{1})$ . Nous construisons également des solutions à logNLS se comportant (dans $L^{2}$ ) comme une somme de N solutions gaussiennes avec différentes vitesses (que nous appelons multi-gaussiennes). Pour chaque cas, la convergence (pour $t \to \infty$ ) est plus rapide qu'exponentielle. Nous prouvons également un résultat de rigidité sur ces multi-gaussiennes et multi-solitons construits, en montrant que ce sont les seuls avec une telle convergence.

We consider the logarithmic Schrödinger equation (logNLS) in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson - a time-independent Gaussian function - is an orbitally stable solution. In this paper, we construct multi-solitons (or multi-Gaussons) for logNLS, with estimates in $H^{1} \cap F (H^{1})$ . We also construct solutions to logNLS behaving (in $L^{2}$ ) like a sum of N Gaussian solutions with different speeds (which we call multi-gaussian). In both cases, the convergence (as $t \to \infty$ ) is faster than exponential. We also prove a rigidity result on these constructed multi-gaussians and multi-solitons, showing that they are the only ones with such a convergence.

Reçu le : 2020-04-10
Révisé le : 2020-09-07
Accepté le : 2020-09-07

DOI : 10.1016/j.anihpc.2020.09.002

Mots-clés : Logarithmic Non-Linear Schrödinger Equation, Multi-soliton, Multi-gaussians, Asymptotic behavior

Affiliations des auteurs :

Ferriere, Guillaume ¹

¹ IMAG, Univ Montpellier, CNRS, Case courrier 051, Place Eugène Bataillon, 34090 Montpellier, France

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     author = {Ferriere, Guillaume},
     title = {Existence of multi-solitons for the focusing {Logarithmic} {Non-Linear} {Schr\"odinger} {Equation}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {841--875},
     publisher = {Elsevier},
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     year = {2021},
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     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2020.09.002/}
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Ferriere, Guillaume. Existence of multi-solitons for the focusing Logarithmic Non-Linear Schrödinger Equation. Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 841-875. doi : 10.1016/j.anihpc.2020.09.002. http://www.numdam.org/articles/10.1016/j.anihpc.2020.09.002/

Bibliographie
Cité par

[1] Ardila, A.H. Orbital stability of Gausson solutions to logarithmic Schrödinger equations, Electron. J. Differ. Equ. (2016) | MR

[2] Bao, W.; Carles, R.; Su, C.; Tang, Q. Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation, SIAM J. Numer. Anal., Volume 57 (2019) no. 2, pp. 657-680 | DOI | MR

[3] Berestycki, H.; Gallouët, T.; Kavian, O. Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris, Ser. I, Volume 297 (1983), pp. 307-310 | MR | Zbl

[4] Berestycki, H.; Lions, P.-L. Nonlinear scalar field equations. II: Existence of infinitely many solutions, Arch. Ration. Mech. Anal., Volume 82 (1983), pp. 347-375 | DOI | MR | Zbl

[5] Białynicki-Birula, I.; Mycielski, J. Nonlinear wave mechanics, Ann. Phys., Volume 100 (1976) no. 1–2, pp. 62-93 | DOI | MR

[6] Buljan, H.; Šiber, A.; Soljačić, M.; Schwartz, T.; Segev, M.; Christodoulides, D.N. Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E (3), Volume 68 (2003) no. 3 | DOI | MR

[7] Carles, R.; Gallagher, I. Universal dynamics for the defocusing logarithmic Schrodinger equation, Duke Math. J., Volume 167 (2018) no. 9, pp. 1761-1801 | DOI | MR

[8] Cazenave, T. Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., Volume 7 (1983) no. 10, pp. 1127-1140 | DOI | MR | Zbl

[9] Cazenave, T. Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences/American Mathematical Society, New York/Providence, RI, 2003 | MR | Zbl

[10] Cazenavec, T.; Haraux, A. Equations d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse (5), Volume II (1980), pp. 21-55 | DOI | Numdam | MR | Zbl

[11] Côte, R. On the soliton resolution for equivariant wave maps to the sphere, Commun. Pure Appl. Math., Volume 68 (2015) no. 11, pp. 1946-2004 | DOI | MR

[12] Côte, R.; Le Coz, S. High-speed excited multi-solitons in nonlinear Schrödinger equations, J. Math. Pures Appl. (9), Volume 96 (2011) no. 2, pp. 135-166 | DOI | MR | Zbl

[13] Côte, R.; Martel, Y. Multi-travelling waves for the nonlinear Klein-Gordon equation, Trans. Am. Math. Soc., Volume 370 (2018) no. 10, pp. 7461-7487 | DOI | MR

[14] Côte, R.; Martel, Y.; Merle, F. Construction of multi-soliton solutions for the $L^{2}$ -supercritical gKdV and NLS equations, Rev. Mat. Iberoam., Volume 27 (2011) no. 1, pp. 273-302 | DOI | MR | Zbl

[15] Côte, R.; Muñoz, C. Multi-solitons for nonlinear Klein-Gordon equations, Forum Math. Sigma, Volume 2 (2014) | DOI | MR | Zbl

[16] d'Avenia, P.; Montefusco, E.; Squassina, M. On the logarithmic Schrödinger equation, Commun. Contemp. Math., Volume 16 (2014) no. 2 | DOI | MR | Zbl

[17] De Martino, S.; Falanga, M.; Godano, C.; Lauro, G. Logarithmic Schrödinger-like equation as a model for magma transport, Europhys. Lett., Volume 63 ( Aug 2003 ) no. 3, pp. 472-475 | DOI

[18] Duyckaerts, T.; Kenig, C.; Merle, F. Classification of the radial solutions of the focusing, energy-critical wave equation, Camb. J. Math., Volume 1 (2013) no. 1, pp. 75-144 | DOI | MR | Zbl

[19] Eckhaus, W.; Schuur, P.C. The emergence of solitons of the Korteweg-de Vries equation from arbitrary initial conditions, Math. Methods Appl. Sci., Volume 5 (1983), pp. 97-116 | DOI | MR | Zbl

[20] Ferriere, G. Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation, Anal. PDE (2020) (to appear) | arXiv | MR

[21] Ferriere, G. The focusing logarithmic Schrödinger equation: analysis of breathers and nonlinear superposition, Discrete Contin. Dyn. Syst., Volume 40 ( Nov 2020 ) no. 11 | DOI | MR

[22] Grillakis, M.; Shatah, J.; Strauss, W. Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., Volume 74 (1987), pp. 160-197 | DOI | MR | Zbl

[23] Grillakis, M.G.; Shatah, J.; Strauss, W. Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal., Volume 94 (1990) no. 2, pp. 308-348 | DOI | MR | Zbl

[24] Guerrero, P.; López, J.L.; Nieto, J. Global $H^{1}$ solvability of the 3D logarithmic Schrödinger equation, Nonlinear Anal., Real World Appl., Volume 11 (2010) no. 1, pp. 79-87 | DOI | MR | Zbl

[25] Hefter, E.F. Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev. A, Volume 32 ( Aug 1985 ), pp. 1201-1204 | DOI

[26] Hernández, E.; Remaud, B. General properties of Gausson-conserving descriptions of quantal damped motion, Phys. A, Stat. Mech. Appl., Volume 105 (1981) no. 1, pp. 130-146 | DOI | MR

[27] Królikowski, W.; Edmundson, D.; Bang, O. Unified model for partially coherent solitons in logarithmically nonlinear media, Phys. Rev. E, Volume 61 ( Mar 2000 ), pp. 3122-3126 | DOI

[28] Le Coz, S. Standing waves in nonlinear Schrödinger equations, Berlin, Germany, 2007–2009, Walter de Gruyter, Berlin (2009), pp. 151-192 | MR | Zbl

[29] Martel, Y. Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Am. J. Math., Volume 127 (2005) no. 5, pp. 1103-1140 | DOI | MR | Zbl

[30] Martel, Y.; Merle, F. Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 23 (2006) no. 6, pp. 849-864 | DOI | Numdam | MR | Zbl

[31] Martel, Y.; Merle, F.; Tsai, T.-P. Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations, Commun. Math. Phys., Volume 231 (2002) no. 2, pp. 347-373 | DOI | MR | Zbl

[32] Martel, Y.; Merle, F.; Tsai, T.-P. Stability in $H^{1}$ of the sum of K solitary waves for some nonlinear Schrödinger equations, Duke Math. J., Volume 133 (2006) no. 3, pp. 405-466 | DOI | MR | Zbl

[33] Merle, F. 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 | DOI | MR | Zbl

[34] Reed, M.; Simon, B. Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978 | MR | Zbl

[35] Schuur, P.C. Asymptotic Analysis of Soliton Problems. An Inverse Scattering Approach, vol. 1232, Springer, Cham, 1986 | MR | Zbl

[36] Troy, W.C. Uniqueness of positive ground state solutions of the logarithmic Schrödinger equation, Arch. Ration. Mech. Anal., Volume 222 (2016) no. 3, pp. 1581-1600 | DOI | MR

[37] Weinstein, M.I. Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., Volume 16 (1985), pp. 472-491 | DOI | MR | Zbl

[38] Weinstein, M.I. Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math., Volume 39 (1986), pp. 51-67 | DOI | MR | Zbl

[39] Zakharov, V.E.; Shabat, A.B. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Zh. Èksp. Teor. Fiz., Volume 61 (1971) no. 1, pp. 118-134 | MR

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