# American Institute of Mathematical Sciences

January  2017, 37(1): 295-336. doi: 10.3934/dcds.2017013

## Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations

 1 Institute of Mathematics, Academia Sinica, Taipei, Taiwan 2 Department of Mathematics, University of British Columbia, Vancouver BC Canada V6T 1Z2, Canada 3 Center for Advanced Study in Theoretical Sciences, National Taiwan University, Taipei, Taiwan

Received  August 2015 Revised  August 2016 Published  November 2016

Fund Project: Lin is currently a postdoctor at Department of Mathematics, National Cheng Kung University, Tainan, Taiwan. Tsai’s research is supported in part by NSERC grant 261356-13.

In this note we construct mixed dimensional infinite soliton trains, which are solutions of nonlinear Schrödinger equations whose asymptotic profiles at time infinity consist of infinitely many solitons of multiple dimensions. For example infinite line-point soliton trains in 2D space, and infinite plane-line-point soliton trains in 3D space. This note extends the works of Le Coz, Li and Tsai [6,7], where single dimensional trains are considered. In our approach, spatial L bounds for lower dimensional trains play an essential role.

Citation: Liren Lin, Tai-Peng Tsai. Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 295-336. doi: 10.3934/dcds.2017013
##### References:
 [1] 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. [2] 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. [3] R. Côte, Y. Martel and F. Merle, Construction of multi-soliton solutions for the L2-supercritical gKdV and NLS equations, Rev. Mat. Iberoam., 27 (2011), 273-302.  doi: 10.4171/RMI/636. [4] P. Deift and J. Park, Long-time asymptotics for solutions of the {NLS} equation with a delta potential and even initial data, Int. Math. Res. Not. IMRN, (2011), 5505-5624.  doi: 10.1007/s11005-010-0458-5. [5] S. Kamvissis, Focusing nonlinear Schrödinger equation with infinitely many solitons, J. Math. Phys., 36 (1995), 4175-4180.  doi: 10.1063/1.530953. [6] S. Le Coz, D. Li and T.-P. Tsai, Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1251-1282.  doi: 10.1017/S030821051500030X. [7] S. Le Coz and T.-P. Tsai, Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations, Nonlinearity, 27 (2014), 2689-2709.  doi: 10.1088/0951-7715/27/11/2689. [8] S. Le Coz and T. -P. Tsai, Finite and infinite soliton and kink-soliton trains of nonlinear Schrödinger equations, To appear in the proceedings of ICCM Ⅵ (Taipei 2013), arXiv: 1409.8379. [9] 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. [10] Y. Martel, F. Merle and T.-P. Tsai, Stability in H1 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. [11] 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. [12] G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Comm. Partial Differential Equations, 29 (2004), 1051-1095.  doi: 10.1081/PDE-200033754. [13] I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of N-soliton states of NLS, ArXiv Mathematics e-prints, math/0309114. [14] V.E. 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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##### References:
 [1] 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. [2] 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. [3] R. Côte, Y. Martel and F. Merle, Construction of multi-soliton solutions for the L2-supercritical gKdV and NLS equations, Rev. Mat. Iberoam., 27 (2011), 273-302.  doi: 10.4171/RMI/636. [4] P. Deift and J. Park, Long-time asymptotics for solutions of the {NLS} equation with a delta potential and even initial data, Int. Math. Res. Not. IMRN, (2011), 5505-5624.  doi: 10.1007/s11005-010-0458-5. [5] S. Kamvissis, Focusing nonlinear Schrödinger equation with infinitely many solitons, J. Math. Phys., 36 (1995), 4175-4180.  doi: 10.1063/1.530953. [6] S. Le Coz, D. Li and T.-P. Tsai, Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1251-1282.  doi: 10.1017/S030821051500030X. [7] S. Le Coz and T.-P. Tsai, Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations, Nonlinearity, 27 (2014), 2689-2709.  doi: 10.1088/0951-7715/27/11/2689. [8] S. Le Coz and T. -P. Tsai, Finite and infinite soliton and kink-soliton trains of nonlinear Schrödinger equations, To appear in the proceedings of ICCM Ⅵ (Taipei 2013), arXiv: 1409.8379. [9] 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. [10] Y. Martel, F. Merle and T.-P. Tsai, Stability in H1 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. [11] 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. [12] G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Comm. Partial Differential Equations, 29 (2004), 1051-1095.  doi: 10.1081/PDE-200033754. [13] I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of N-soliton states of NLS, ArXiv Mathematics e-prints, math/0309114. [14] V.E. 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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