no. 3
Existence of multi-solitons for the focusing Logarithmic Non-Linear Schrödinger Equation
Ferriere, Guillaume1
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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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 H1∩F(H1). We also construct solutions to logNLS behaving (in L2) like a sum of N Gaussian solutions with different speeds (which we call multi-gaussian). In both cases, the convergence (as t→∞) 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.

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 H1∩F(H1). Nous construisons également des solutions à logNLS se comportant (dans L2) comme une somme de N solutions gaussiennes avec différentes vitesses (que nous appelons multi-gaussiennes). Pour chaque cas, la convergence (pour t→∞) 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.

Reçu le :
Révisé le :
Accepté le :
DOI : 10.1016/j.anihpc.2020.09.002
Keywords: Logarithmic Non-Linear Schrödinger Equation, Multi-soliton, Multi-gaussians, Asymptotic behavior

Ferriere, Guillaume 1

1 IMAG, Univ Montpellier, CNRS, Case courrier 051, Place Eugène Bataillon, 34090 Montpellier, France 
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     title = {Existence of multi-solitons for the focusing {Logarithmic} {Non-Linear} {Schr\"odinger} {Equation}},
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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

[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 | MR | DOI

[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 | MR | Zbl | DOI

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

[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 | MR | DOI

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

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

[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 | MR | Zbl | Numdam | DOI

[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 | MR | DOI

[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 | MR | Zbl | DOI

[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 | MR | DOI

[14] Côte, R.; Martel, Y.; Merle, F. Construction of multi-soliton solutions for the L2-supercritical gKdV and NLS equations, Rev. Mat. Iberoam., Volume 27 (2011) no. 1, pp. 273-302 | MR | Zbl | DOI

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

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

[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 | MR | Zbl | DOI

[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 | MR | Zbl | DOI

[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 | MR | DOI

[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 | MR | Zbl | DOI

[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 | MR | Zbl | DOI

[24] Guerrero, P.; López, J.L.; Nieto, J. Global H1 solvability of the 3D logarithmic Schrödinger equation, Nonlinear Anal., Real World Appl., Volume 11 (2010) no. 1, pp. 79-87 | MR | Zbl | DOI

[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 | MR | DOI

[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 | MR | Zbl | DOI

[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 | MR | Zbl | Numdam | DOI

[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 | MR | Zbl | DOI

[32] Martel, Y.; Merle, F.; Tsai, T.-P. Stability in H1 of the sum of K solitary waves for some nonlinear Schrödinger equations, Duke Math. J., Volume 133 (2006) no. 3, pp. 405-466 | MR | Zbl | DOI

[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 | MR | Zbl | DOI

[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 | MR | DOI

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

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

[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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