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March  2020, 40(3): 1595-1620. doi: 10.3934/dcds.2020087

## Construction of 2-solitons with logarithmic distance for the one-dimensional cubic Schrödinger system

 CMLS, École Polytechnique, CNRS, 91128 Palaiseau, France

* Corresponding author: Yvan Martel

Received  March 2019 Published  December 2019

We consider a system of coupled cubic Schrödinger equations in one space dimension
 $\begin{equation*} \begin{cases} \text{i}\partial_t u + \partial_x^2 u +(|u|^2 + \omega |v|^2) u = 0\\ \text{i}\partial_t v + \partial_x^2 v+ (|v|^2 + \omega |u|^2) v = 0 \end{cases}\quad (t,x)\in \mathbb{R}\times \mathbb{R}, \end{equation*}$
in the non-integrable case
 $0 < \omega < 1$
.
First, we justify the existence of a symmetric 2-solitary wave with logarithmic distance, i.e. a solution of the system satisfying
 $\lim\limits_{t\to +\infty}\left\| \begin{pmatrix} u(t) \\ v(t)\end{pmatrix} - \begin{pmatrix} e^{ \text{i} t}Q (\cdot - \frac{1}{2} \log (\Omega t) - \frac{1}{4} \log \log t) \\[4pt] e^{ \text{i} t}Q (\cdot + \frac{1}{2} \log (\Omega t) + \frac{1}{4} \log \log t)\end{pmatrix}\right\|_{H^1\times H^1} = 0$
where
 $Q = \sqrt{2}$
sech is the explicit solution of
 $Q'' - Q + Q^3 = 0$
and
 $\Omega>0$
is a constant. This result extends to the non-integrable case the existence of symmetric 2-solitons with logarithmic distance known in the integrable case
 $\omega = 0$
and
 $\omega = 1$
([15,33]). Such strongly interacting symmetric
 $2$
-solitary waves were also previously constructed for the non-integrable scalar nonlinear Schrödinger equation in any space dimension and for any energy-subcritical power nonlinearity ([20,22]).
Second, under the conditions
 $0 and $ 0<\omega < \frac 12 c(c+1) $, we construct solutions of the system satisfying $ \lim\limits_{t\to +\infty}\left\| \begin{pmatrix}u(t) \\ v(t)\end{pmatrix} - \begin{pmatrix}e^{ \text{i} c^2 t}Q_c (\cdot - \frac{1}{(c+1)c} \log (\Omega_c t) )\\[4pt] e^{ \text{i} t} Q (\cdot + \frac{1}{c+1} \log (\Omega_c t))\end{pmatrix} \right\|_{H^1\times H^1} = 0 $where $ Q_c(x) = cQ(cx) $and $ \Omega_c>0 $is a constant. Such logarithmic regime with non-symmetric solitons does not exist in the integrable cases $ \omega = 0 $and $ \omega = 1 $and is still unknown in the non-integrable scalar case. Citation: Yvan Martel, Tiễn Vinh Nguyến. Construction of 2-solitons with logarithmic distance for the one-dimensional cubic Schrödinger system. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1595-1620. doi: 10.3934/dcds.2020087 ##### References:  [1] M. J. Ablowitz, B. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, London Mathematical Society Lecture Note Series, 302, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511546709. [2] L. Bergé, Wave collapse in physics: Principles and applications to light and plasma waves, Phys. Rep., 303 (1998), 259-370. doi: 10.1016/S0370-1573(97)00092-6. [3] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. [4] V. Combet and Y. Martel, Construction of multi-bubble solutions for the critical GKDV equation, SIAM J. Math. Anal., 50 (2018), 3715-3790. doi: 10.1137/17M1140595. [5] R. Côte, Y. Martel and F. Merle, Construction of multi-soliton solutions for the$L^2$-supercritical gKdV and NLS equations, Rev. Mat. Iberoam., 27 (2011), 273-302. doi: 10.4171/RMI/636. [6] F. Delebecque, S. Le Coz and R. M. Weishäupl, Multi-speed solitary waves of nonlinear Schrödinger systems: Theoretical and numerical analysis, Commun. Math. Sci., 14 (2016), 1599-1624. doi: 10.4310/CMS.2016.v14.n6.a7. [7] L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Classics in Mathematics, Springer, Berlin, 2007. [8] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Functional Analysis, 32 (1979), 1-32. doi: 10.1016/0022-1236(79)90076-4. [9] K. A. Gorshkov and L. A. Ostrovsky, Interactions of solitons in non-integrable systems: Direct perturbation method and applications, Physica 3D, 1 & 2 (1981), 428-438. [10] M. Grillakis, J. Shatah and W. A. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9. [11] I. Ianni and S. Le Coz, Multi-speed solitary wave solutions for nonlinear Schrödinger system, J. Lond. Math. Soc. (2), 89 (2014), 623-639. doi: 10.1112/jlms/jdt083. [12] J. Jendrej, Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations, preprint, arXiv: 1802.06294 [13] V. I. Karpman and V. V. Solov'ev, A perturbational approach to the two-soliton system, Phys. D, 3 (1981), 487-502. doi: 10.1016/0167-2789(81)90035-X. [14] J. Krieger, Y. Martel and P. Raphaël, Two-soliton solutions to the three-dimensional gravitational Hartree equation, Comm. Pure Appl. Math., 62 (2009), 1501-1550. doi: 10.1002/cpa.20292. [15] S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, J. Experimental Theoretical Physics, 38 (1974), 248-253. [16] Y. Martel, Asymptotic$N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math., 127 (2005), 1103-1140. doi: 10.1353/ajm.2005.0033. [17] Y. Martel and F. Merle, Multi-solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 849-864. doi: 10.1016/j.anihpc.2006.01.001. [18] Y. Martel, F. Merle and T.-P. Tsai, Stability and asymptotic stability in the energy space of the sum of$N$solitons for the subcritical gKdV equations, Commun. Math. Phys., 231 (2002), 347-373. doi: 10.1007/s00220-002-0723-2. [19] Y. Martel, F. Merle and T.-P. Tsai, Stability in$H^1$of the sum of$K$solitary waves for some nonlinear Schrödinger equations, Duke Math. J., 133 (2006), 405-466. doi: 10.1215/S0012-7094-06-13331-8. [20] Y. Martel and P. Raphaël, Strongly interacting blow up bubbles for the mass critical nonlinear Schrödinger equation, Ann. Sci. Éc. Norm. Supér. (4), 51 (2018), 701–737. doi: 10.24033/asens.2364. [21] F. Merle, Construction of solutions with exactly$k$blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., 129 (1990), 223-240. doi: 10.1007/BF02096981. [22] T. V. Nguyễn, Existence of multi-solitary waves with logarithmic relative distances for the NLS equations, C. R. Math. Acad. Sci. Paris, 357 (2019), 13-58. doi: 10.1016/j.crma.2018.11.012. [23] T. V. Nguyễn, Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation, Nonlinearity, 30 (2017), 4614-4648. doi: 10.1088/1361-6544/aa8cab. [24] E. Olmedilla, Multiple pole solutions of the nonlinear Schrödinger equation, Phys. D, 25 (1987), 330-346. doi: 10.1016/0167-2789(87)90107-2. [25] P. Raphaël, Stability and blow up for the nonlinear Schrödinger equation, in Lecture Notes for the Clay Summer School on Evolution Equations, ETH, Zurich, 2008. [26] P. Raphaël and J. Szeftel, Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc., 24 (2011), 471-546. doi: 10.1090/S0894-0347-2010-00688-1. [27] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Clarendon Press, Oxford, 1946. [28] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. doi: 10.1007/BF01208265. [29] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034. [30] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67. doi: 10.1002/cpa.3160390103. [31] J. Yang, Suppression of Manakov-soliton interference in optical fibers, Rev. E., 65 (2002). doi: 10.1103/PhysRevE.65.036606. [32] J. Yang, Nonlinear Waves in Integrable and Non-Integrable Systems, Mathematical Modeling and Computation, 16, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898719680. [33] T. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), 62-69. show all references ##### References:  [1] M. J. Ablowitz, B. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, London Mathematical Society Lecture Note Series, 302, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511546709. [2] L. Bergé, Wave collapse in physics: Principles and applications to light and plasma waves, Phys. Rep., 303 (1998), 259-370. doi: 10.1016/S0370-1573(97)00092-6. [3] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. [4] V. Combet and Y. Martel, Construction of multi-bubble solutions for the critical GKDV equation, SIAM J. Math. Anal., 50 (2018), 3715-3790. doi: 10.1137/17M1140595. [5] R. Côte, Y. Martel and F. Merle, Construction of multi-soliton solutions for the$L^2$-supercritical gKdV and NLS equations, Rev. Mat. Iberoam., 27 (2011), 273-302. doi: 10.4171/RMI/636. [6] F. Delebecque, S. Le Coz and R. M. Weishäupl, Multi-speed solitary waves of nonlinear Schrödinger systems: Theoretical and numerical analysis, Commun. Math. Sci., 14 (2016), 1599-1624. doi: 10.4310/CMS.2016.v14.n6.a7. [7] L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Classics in Mathematics, Springer, Berlin, 2007. [8] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Functional Analysis, 32 (1979), 1-32. doi: 10.1016/0022-1236(79)90076-4. [9] K. A. Gorshkov and L. A. Ostrovsky, Interactions of solitons in non-integrable systems: Direct perturbation method and applications, Physica 3D, 1 & 2 (1981), 428-438. [10] M. Grillakis, J. Shatah and W. A. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9. [11] I. Ianni and S. Le Coz, Multi-speed solitary wave solutions for nonlinear Schrödinger system, J. Lond. Math. Soc. (2), 89 (2014), 623-639. doi: 10.1112/jlms/jdt083. [12] J. Jendrej, Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations, preprint, arXiv: 1802.06294 [13] V. I. Karpman and V. V. Solov'ev, A perturbational approach to the two-soliton system, Phys. D, 3 (1981), 487-502. doi: 10.1016/0167-2789(81)90035-X. [14] J. Krieger, Y. Martel and P. Raphaël, Two-soliton solutions to the three-dimensional gravitational Hartree equation, Comm. Pure Appl. Math., 62 (2009), 1501-1550. doi: 10.1002/cpa.20292. [15] S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, J. Experimental Theoretical Physics, 38 (1974), 248-253. [16] Y. Martel, Asymptotic$N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math., 127 (2005), 1103-1140. doi: 10.1353/ajm.2005.0033. [17] Y. Martel and F. Merle, Multi-solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 849-864. doi: 10.1016/j.anihpc.2006.01.001. [18] Y. Martel, F. Merle and T.-P. Tsai, Stability and asymptotic stability in the energy space of the sum of$N$solitons for the subcritical gKdV equations, Commun. Math. Phys., 231 (2002), 347-373. doi: 10.1007/s00220-002-0723-2. [19] Y. Martel, F. Merle and T.-P. Tsai, Stability in$H^1$of the sum of$K$solitary waves for some nonlinear Schrödinger equations, Duke Math. J., 133 (2006), 405-466. doi: 10.1215/S0012-7094-06-13331-8. [20] Y. Martel and P. Raphaël, Strongly interacting blow up bubbles for the mass critical nonlinear Schrödinger equation, Ann. Sci. Éc. Norm. Supér. (4), 51 (2018), 701–737. doi: 10.24033/asens.2364. [21] F. Merle, Construction of solutions with exactly$k\$ blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., 129 (1990), 223-240.  doi: 10.1007/BF02096981. [22] T. V. Nguyễn, Existence of multi-solitary waves with logarithmic relative distances for the NLS equations, C. R. Math. Acad. Sci. Paris, 357 (2019), 13-58.  doi: 10.1016/j.crma.2018.11.012. [23] T. V. Nguyễn, Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation, Nonlinearity, 30 (2017), 4614-4648.  doi: 10.1088/1361-6544/aa8cab. [24] E. Olmedilla, Multiple pole solutions of the nonlinear Schrödinger equation, Phys. D, 25 (1987), 330-346.  doi: 10.1016/0167-2789(87)90107-2. [25] P. Raphaël, Stability and blow up for the nonlinear Schrödinger equation, in Lecture Notes for the Clay Summer School on Evolution Equations, ETH, Zurich, 2008. [26] P. Raphaël and J. Szeftel, Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc., 24 (2011), 471-546.  doi: 10.1090/S0894-0347-2010-00688-1. [27] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Clarendon Press, Oxford, 1946. [28] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.  doi: 10.1007/BF01208265. [29] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034. [30] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67.  doi: 10.1002/cpa.3160390103. [31] J. Yang, Suppression of Manakov-soliton interference in optical fibers, Rev. E., 65 (2002). doi: 10.1103/PhysRevE.65.036606. [32] J. Yang, Nonlinear Waves in Integrable and Non-Integrable Systems, Mathematical Modeling and Computation, 16, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898719680. [33] T. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), 62-69.
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