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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[3]

R. CôteY. 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.  Google Scholar

[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.  Google Scholar

[5]

S. Kamvissis, Focusing nonlinear Schrödinger equation with infinitely many solitons, J. Math. Phys., 36 (1995), 4175-4180.  doi: 10.1063/1.530953.  Google Scholar

[6]

S. Le CozD. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[10]

Y. MartelF. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of N-soliton states of NLS, ArXiv Mathematics e-prints, math/0309114. Google Scholar

[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.   Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

R. CôteY. 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.  Google Scholar

[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.  Google Scholar

[5]

S. Kamvissis, Focusing nonlinear Schrödinger equation with infinitely many solitons, J. Math. Phys., 36 (1995), 4175-4180.  doi: 10.1063/1.530953.  Google Scholar

[6]

S. Le CozD. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[10]

Y. MartelF. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of N-soliton states of NLS, ArXiv Mathematics e-prints, math/0309114. Google Scholar

[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.   Google Scholar

[1]

Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379

[2]

W. Josh Sonnier, C. I. Christov. Repelling soliton collisions in coupled Schrödinger equations with negative cross modulation. Conference Publications, 2009, 2009 (Special) : 708-718. doi: 10.3934/proc.2009.2009.708

[3]

Marius Ghergu, Gurpreet Singh. On a class of mixed Choquard-Schrödinger-Poisson systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 297-309. doi: 10.3934/dcdss.2019021

[4]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[5]

Piotr Gwiazda, Karolina Kropielnicka, Anna Marciniak-Czochra. The Escalator Boxcar Train method for a system of age-structured equations. Networks & Heterogeneous Media, 2016, 11 (1) : 123-143. doi: 10.3934/nhm.2016.11.123

[6]

Georg Vossen, Torsten Hermanns. On an optimal control problem in laser cutting with mixed finite-/infinite-dimensional constraints. Journal of Industrial & Management Optimization, 2014, 10 (2) : 503-519. doi: 10.3934/jimo.2014.10.503

[7]

Yi He, Gongbao Li. Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 731-762. doi: 10.3934/dcds.2016.36.731

[8]

Abbas Moameni. Soliton solutions for quasilinear Schrödinger equations involving supercritical exponent in $\mathbb R^N$. Communications on Pure & Applied Analysis, 2008, 7 (1) : 89-105. doi: 10.3934/cpaa.2008.7.89

[9]

Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 933-982. doi: 10.3934/dcds.2020067

[10]

Toshiyuki Suzuki. Nonlinear Schrödinger equations with inverse-square potentials in two dimensional space. Conference Publications, 2015, 2015 (special) : 1019-1024. doi: 10.3934/proc.2015.1019

[11]

Chengchun Hao. Well-posedness for one-dimensional derivative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 997-1021. doi: 10.3934/cpaa.2007.6.997

[12]

Antonio Di Crescenzo, Maria Longobardi, Barbara Martinucci. On a spike train probability model with interacting neural units. Mathematical Biosciences & Engineering, 2014, 11 (2) : 217-231. doi: 10.3934/mbe.2014.11.217

[13]

Joseph Bayara, André Conseibo, Moussa Ouattara, Artibano Micali. Train algebras of degree 2 and exponent 3. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1371-1386. doi: 10.3934/dcdss.2011.4.1371

[14]

Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337

[15]

Alexander Pankov. Nonlinear Schrödinger Equations on Periodic Metric Graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 697-714. doi: 10.3934/dcds.2018030

[16]

Guoyuan Chen, Youquan Zheng. Concentration phenomenon for fractional nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2359-2376. doi: 10.3934/cpaa.2014.13.2359

[17]

Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565

[18]

Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703

[19]

Nobu Kishimoto. A remark on norm inflation for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1375-1402. doi: 10.3934/cpaa.2019067

[20]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (19)
  • HTML views (9)
  • Cited by (0)

Other articles
by authors

[Back to Top]