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November  2020, 40(11): 6247-6274. doi: 10.3934/dcds.2020277

The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition

Guillaume Ferriere

IMAG, Univ. Montpellier, CNRS, Montpellier, France

Received  November 2019 Revised  May 2020 Published  July 2020

We consider the logarithmic Schrödinger equation 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 the general case in dimension $ d = 1 $, the solution with Gaussian initial data is periodic, and we compute some approximations of the period in the case of small and large oscillations, showing that the period can be as large as wanted for the latter. The main result of this article is a principle of nonlinear superposition: starting from an initial data made of the sum of several standing Gaussian functions far from each other, the solution remains close (in $ L^2 $) to the sum of the corresponding Gaussian solutions for a long time, in square of the distance between the Gaussian functions.

Keywords: Logarithmic Schrödinger equation, Gaussons, breathers, Gaussian solitary waves, nonlinear dynamics, nonlinear superposition.
Mathematics Subject Classification: Primary: 35Q55, 35B10, 35B15, 35B40; Secondary: 34C25, 34C27.
Citation: Guillaume Ferriere. The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6247-6274. doi: 10.3934/dcds.2020277
References:
[1]

M. A. Alejo and C. Muñoz, Nonlinear stability of MKdV breathers, Comm. Math. Phys., 324 (2013), 233-262.  doi: 10.1007/s00220-013-1792-0.  Google Scholar

[2]

M. A. Alejo and C. Muñoz, Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers, Anal. PDE, 8 (2015), 629-674.  doi: 10.2140/apde.2015.8.629.  Google Scholar

[3]

A. H. Ardila, Orbital stability of Gausson solutions to logarithmic Schrödinger equations, Electron. J. Differential Equations, (2016), Paper No. 335, 9pp.  Google Scholar

[4]

W. Bao, R. Carles, C. Su and Q. Tang, Error estimates of a regularized finite difference method for the logarithmic {S}chrödinger equation, SIAM J. Numer. Anal., 57 (2019), 657-680.  doi: 10.1137/18M1177445.  Google Scholar

[5]

W. Bao, R. Carles, C. Su and Q. Tang, Regularized numerical methods for the logarithmic Schrödinger equation, Numer. Math., 143 (2019), 461-487.  doi: 10.1007/s00211-019-01058-2.  Google Scholar

[6]

I. Białynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics, 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.  Google Scholar

[7]

H. Buljan, A. Šiber, M. Soljačić, T. Schwartz, M. Segev and D. N. Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E (3), 68 (2003), 036607, 6pp. doi: 10.1103/PhysRevE.68.036607.  Google Scholar

[8]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic Schrodinger equation, Duke Mathematical Journal, 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.  Google Scholar

[9]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., 7 (1983), 1127-1140.  doi: 10.1016/0362-546X(83)90022-6.  Google Scholar

[10]

T. Cazenave, Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[11]

T. Cazenave and A. Haraux, Équations d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math. (5), 2 (1980), 21–51. doi: 10.5802/afst.543.  Google Scholar

[12]

S.-M. Chang, S. Gustafson, K. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal., 39 (2007/08), 1070-1111.  doi: 10.1137/050648389.  Google Scholar

[13]

G. Chen and J. Liu, Soliton resolution for the modified KdV equation, 2019, arXiv: 1907.07115, Preprint. Google Scholar

[14]

R. Côte and S. Le Coz, High-speed excited multi-solitons in nonlinear Schrödinger equations, J. Math. Pures Appl. (9), 96 (2011), 135–166. doi: 10.1016/j.matpur.2011.03.004.  Google Scholar

[15]

R. Côte, Y. Martel and F. Merle, Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations, Rev. Mat. Iberoam., 27 (2011), 273-302.   Google Scholar

[16]

P. d'Avenia, E. Montefusco and M. Squassina, On the logarithmic {S}chrödinger equation, Commun. Contemp. Math., 16 (2014), 1350032, 15pp. doi: 10.1142/S0219199713500326.  Google Scholar

[17]

S. De Martino, M. Falanga, C. Godano and G. Lauro, Logarithmic schrödinger-like equation as a model for magma transport, EPL (Europhysics Letters), 63 (2003), 472-475.  doi: 10.1209/epl/i2003-00547-6.  Google Scholar

[18]

G. Ferriere, Existence of multi-solitons for the focusing logarithmic Schrödinger equation, arXiv: 2003.02571, Preprint. Google Scholar

[19]

G. Ferriere, Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation, Analysis & PDE, to appear. Google Scholar

[20]

M. Grillakis, Existence of nodal solutions of semilinear equations in ${\bf{R}}^N$, J. Differential Equations, 85 (1990), 367-400.  doi: 10.1016/0022-0396(90)90121-5.  Google Scholar

[21]

M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, Comm. Pure Appl. Math., 41 (1988), 747-774.  doi: 10.1002/cpa.3160410602.  Google Scholar

[22]

P. Guerrero, J. L. López and J. Nieto, Global $H^1$ solvability of the 3D logarithmic Schrödinger equation, Nonlinear Anal. Real World Appl., 11 (2010), 79-87.  doi: 10.1016/j.nonrwa.2008.10.017.  Google Scholar

[23]

E. F. Hefter, Application of the nonlinear schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev. A, 32 (1985), 1201-1204.  doi: 10.1103/PhysRevA.32.1201.  Google Scholar

[24]

E. Hernández and B. Remaud, General properties of gausson-conserving descriptions of quantal damped motion, Phys. A, 105 (1981), 130-146.  doi: 10.1016/0378-4371(81)90066-2.  Google Scholar

[25]

C. K. R. T. Jones, An instability mechanism for radially symmetric standing waves of a nonlinear {S}chrödinger equation, J. Differential Equations, 71 (1988), 34-62.  doi: 10.1016/0022-0396(88)90037-X.  Google Scholar

[26]

W. Królikowski, D. Edmundson and O. Bang, Unified model for partially coherent solitons in logarithmically nonlinear media, Phys. Rev. E, 61 (2000), 3122-3126.  doi: 10.1103/PhysRevE.61.3122.  Google Scholar

[27]

Y. Martel and F. Merle, Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 849-864.  doi: 10.1016/j.anihpc.2006.01.001.  Google Scholar

[28]

Y. Martel, F. Merle and T.-P. Tsai, Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations, Comm. Math. Phys., 231 (2002), 347-373.  doi: 10.1007/s00220-002-0723-2.  Google Scholar

[29]

Y. Martel, F. Merle and T.-P. Tsai, Stability in $H^1$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations, Duke Math. J., 133 (2006), 405-466.  doi: 10.1215/S0012-7094-06-13331-8.  Google Scholar

[30]

R. M. Miura, The Korteweg-de Vries equation: A survey of results, SIAM Rev., 18 (1976), 412-459.  doi: 10.1137/1018076.  Google Scholar

[31]

T. Mizumachi, Instability of bound states for 2D nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 13 (2005), 413-428.  doi: 10.3934/dcds.2005.13.413.  Google Scholar

[32]

T. Mizumachi, Vortex solitons for 2D focusing nonlinear Schrödinger equation, Differential Integral Equations, 18 (2005), 431-450.   Google Scholar

[33]

T. Mizumachi, Instability of vortex solitons for 2D focusing NLS, Adv. Differential Equations, 12 (2007), 241-264.   Google Scholar

[34]

Z. Opial, Sur les périodes des solutions de l'équation différentielle $x{\prime\prime}+g(x) = 0$, Ann. Polon. Math., 10 (1961), 49-72.  doi: 10.4064/ap-10-1-49-72.  Google Scholar

[35]

W. Walter, Ordinary Differential Equations, vol. 182 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1998, Translated from the sixth German (1996) edition by Russell Thompson, Readings in Mathematics. doi: 10.1007/978-1-4612-0601-9.  Google Scholar

[36]

V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Èksper. Teoret. Fiz., 61 (1971), 118–134.  Google Scholar

show all references

References:
[1]

M. A. Alejo and C. Muñoz, Nonlinear stability of MKdV breathers, Comm. Math. Phys., 324 (2013), 233-262.  doi: 10.1007/s00220-013-1792-0.  Google Scholar

[2]

M. A. Alejo and C. Muñoz, Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers, Anal. PDE, 8 (2015), 629-674.  doi: 10.2140/apde.2015.8.629.  Google Scholar

[3]

A. H. Ardila, Orbital stability of Gausson solutions to logarithmic Schrödinger equations, Electron. J. Differential Equations, (2016), Paper No. 335, 9pp.  Google Scholar

[4]

W. Bao, R. Carles, C. Su and Q. Tang, Error estimates of a regularized finite difference method for the logarithmic {S}chrödinger equation, SIAM J. Numer. Anal., 57 (2019), 657-680.  doi: 10.1137/18M1177445.  Google Scholar

[5]

W. Bao, R. Carles, C. Su and Q. Tang, Regularized numerical methods for the logarithmic Schrödinger equation, Numer. Math., 143 (2019), 461-487.  doi: 10.1007/s00211-019-01058-2.  Google Scholar

[6]

I. Białynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics, 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.  Google Scholar

[7]

H. Buljan, A. Šiber, M. Soljačić, T. Schwartz, M. Segev and D. N. Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E (3), 68 (2003), 036607, 6pp. doi: 10.1103/PhysRevE.68.036607.  Google Scholar

[8]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic Schrodinger equation, Duke Mathematical Journal, 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.  Google Scholar

[9]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., 7 (1983), 1127-1140.  doi: 10.1016/0362-546X(83)90022-6.  Google Scholar

[10]

T. Cazenave, Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[11]

T. Cazenave and A. Haraux, Équations d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math. (5), 2 (1980), 21–51. doi: 10.5802/afst.543.  Google Scholar

[12]

S.-M. Chang, S. Gustafson, K. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal., 39 (2007/08), 1070-1111.  doi: 10.1137/050648389.  Google Scholar

[13]

G. Chen and J. Liu, Soliton resolution for the modified KdV equation, 2019, arXiv: 1907.07115, Preprint. Google Scholar

[14]

R. Côte and S. Le Coz, High-speed excited multi-solitons in nonlinear Schrödinger equations, J. Math. Pures Appl. (9), 96 (2011), 135–166. doi: 10.1016/j.matpur.2011.03.004.  Google Scholar

[15]

R. Côte, Y. Martel and F. Merle, Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations, Rev. Mat. Iberoam., 27 (2011), 273-302.   Google Scholar

[16]

P. d'Avenia, E. Montefusco and M. Squassina, On the logarithmic {S}chrödinger equation, Commun. Contemp. Math., 16 (2014), 1350032, 15pp. doi: 10.1142/S0219199713500326.  Google Scholar

[17]

S. De Martino, M. Falanga, C. Godano and G. Lauro, Logarithmic schrödinger-like equation as a model for magma transport, EPL (Europhysics Letters), 63 (2003), 472-475.  doi: 10.1209/epl/i2003-00547-6.  Google Scholar

[18]

G. Ferriere, Existence of multi-solitons for the focusing logarithmic Schrödinger equation, arXiv: 2003.02571, Preprint. Google Scholar

[19]

G. Ferriere, Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation, Analysis & PDE, to appear. Google Scholar

[20]

M. Grillakis, Existence of nodal solutions of semilinear equations in ${\bf{R}}^N$, J. Differential Equations, 85 (1990), 367-400.  doi: 10.1016/0022-0396(90)90121-5.  Google Scholar

[21]

M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, Comm. Pure Appl. Math., 41 (1988), 747-774.  doi: 10.1002/cpa.3160410602.  Google Scholar

[22]

P. Guerrero, J. L. López and J. Nieto, Global $H^1$ solvability of the 3D logarithmic Schrödinger equation, Nonlinear Anal. Real World Appl., 11 (2010), 79-87.  doi: 10.1016/j.nonrwa.2008.10.017.  Google Scholar

[23]

E. F. Hefter, Application of the nonlinear schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev. A, 32 (1985), 1201-1204.  doi: 10.1103/PhysRevA.32.1201.  Google Scholar

[24]

E. Hernández and B. Remaud, General properties of gausson-conserving descriptions of quantal damped motion, Phys. A, 105 (1981), 130-146.  doi: 10.1016/0378-4371(81)90066-2.  Google Scholar

[25]

C. K. R. T. Jones, An instability mechanism for radially symmetric standing waves of a nonlinear {S}chrödinger equation, J. Differential Equations, 71 (1988), 34-62.  doi: 10.1016/0022-0396(88)90037-X.  Google Scholar

[26]

W. Królikowski, D. Edmundson and O. Bang, Unified model for partially coherent solitons in logarithmically nonlinear media, Phys. Rev. E, 61 (2000), 3122-3126.  doi: 10.1103/PhysRevE.61.3122.  Google Scholar

[27]

Y. Martel and F. Merle, Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 849-864.  doi: 10.1016/j.anihpc.2006.01.001.  Google Scholar

[28]

Y. Martel, F. Merle and T.-P. Tsai, Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations, Comm. Math. Phys., 231 (2002), 347-373.  doi: 10.1007/s00220-002-0723-2.  Google Scholar

[29]

Y. Martel, F. Merle and T.-P. Tsai, Stability in $H^1$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations, Duke Math. J., 133 (2006), 405-466.  doi: 10.1215/S0012-7094-06-13331-8.  Google Scholar

[30]

R. M. Miura, The Korteweg-de Vries equation: A survey of results, SIAM Rev., 18 (1976), 412-459.  doi: 10.1137/1018076.  Google Scholar

[31]

T. Mizumachi, Instability of bound states for 2D nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 13 (2005), 413-428.  doi: 10.3934/dcds.2005.13.413.  Google Scholar

[32]

T. Mizumachi, Vortex solitons for 2D focusing nonlinear Schrödinger equation, Differential Integral Equations, 18 (2005), 431-450.   Google Scholar

[33]

T. Mizumachi, Instability of vortex solitons for 2D focusing NLS, Adv. Differential Equations, 12 (2007), 241-264.   Google Scholar

[34]

Z. Opial, Sur les périodes des solutions de l'équation différentielle $x{\prime\prime}+g(x) = 0$, Ann. Polon. Math., 10 (1961), 49-72.  doi: 10.4064/ap-10-1-49-72.  Google Scholar

[35]

W. Walter, Ordinary Differential Equations, vol. 182 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1998, Translated from the sixth German (1996) edition by Russell Thompson, Readings in Mathematics. doi: 10.1007/978-1-4612-0601-9.  Google Scholar

[36]

V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Èksper. Teoret. Fiz., 61 (1971), 118–134.  Google Scholar

Figure 1.  Phase portraits. At the left-hand side, the arrows give the direction of the flow; the green and magenta lines are the horizontal and vertical isoclines respectively; the red lines are some trajectories, which fits the level set of $ H(p, q) $. At the right-hand side, a part of some trajectories are drawn along with the isoclines again
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Figure 2.  Plot of some trajectories of $ \tau_\gamma $ in the phase space for 5 real $ \gamma $ between $ 10 $ and $ 50 $
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Figure 3.  Plot of the breather ($ \lambda = 0.5 $) with initial data $ \exp \Bigl( \frac{1}{2} - \delta x^2 \Bigr) $ with $ \delta^{-1} = 2 \times 35^2 $. The first "peak" is at $ t = 43.86 $
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