American Institute of Mathematical Sciences

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September  2011, 31(3): 737-752. doi: 10.3934/dcds.2011.31.737

Homoclinic standing waves in focusing DNLS equations

Michael Herrmann 1,

1. 

Universität des Saarlandes, FR Mathematik, Postfach 15 11 50, D-66041 Saarbrücken, Germany

Received  February 2010 Revised  July 2011 Published  August 2011

We study focusing discrete nonlinear Schrödinger equations and present a novel variational existence proof for homoclinic standing waves (bright solitons). Our approach relies on the constrained maximization of an energy functional and provides the existence of two one-parameter families of waves with unimodal and even profile function for a wide class of nonlinearities. Finally, we illustrate our results by numerical simulations.
Keywords: homoclinic standing waves, breathers., bright solitons, Discrete nonlinear Schrödinger equation (DNLS), solitary waves, nonlinear lattice waves.
Mathematics Subject Classification: Primary: 37K60, 47J30; Secondary: 78A4.
Citation: Michael Herrmann. Homoclinic standing waves in focusing DNLS equations. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 737-752. doi: 10.3934/dcds.2011.31.737
References:
[1]

S. Aubry, Breathers in nonlinear lattices: Existence, linear stability and quantization, Lattice Dynamics (Paris, 1995), Physica D, 103 (1997), 201-250. doi: 10.1016/S0167-2789(96)00261-8.

[2]

W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697. doi: 10.1137/S1064827503422956.

[3]

J. Dorignac, J. Zhou and D. K. Campbell, Discrete breathers in nonlinear Schrödinger hypercubic lattices with arbitrary power nonlinearity, Physica D, 237 (2008), 486-504. doi: 10.1016/j.physd.2007.09.018.

[4]

J. Ch. Eilbeck and M. Johansson, The discrete nonlinear Schrödinger equation - 20 years on, in Proceedings of the Third Conference on Localization and Energy Transfer in Nonlinear Systems 2002, 44-87, 2003.

[5]

J. C. Eilbeck, P. S. Lomdahl and A. C. Scott, The discrete self-trapping equation, Physica D, 16 (1985), 318-338. doi: 10.1016/0167-2789(85)90012-0.

[6]

S. Flach, K. Kladko and R. S. MacKay, Energy thresholds for discrete breathers in one-, two-, and three-dimensional lattices, Phys. Rev. Lett., 78 (1987), 1207-1210. doi: 10.1103/PhysRevLett.78.1207.

[7]

M. Herrmann, Heteroclinic standing waves in defocussing DNLS equations: Variational approach via energy minimization, Appl. Anal., 89 (2010), 1591-1602, arXiv:1002.1591.

[8]

P. G. Kevrekidis, "The Discrete Nonlinear Schrödinger Equation," Springer Tracts in Modern Physics, 232, Chapter 1, General Introduction and Derivation of the DNLS Equation, 3-9, Springer, Berlin, 2009.

[9]

P. G. Kevrekidis, "The Discrete Nonlinear Schrödinger Equation," Springer Tracts in Modern Physics, 232, Chapter 5, The Defocusing Case, 117-141. Springer, Berlin, 2009.

[10]

P. G. Kevrekidis, "The Discrete Nonlinear Schrödinger Equation," Springer Tracts in Modern Physics, 232, Chapter 2, The One-Dimensional Case, 3-9, Springer, Berlin, 2009.

[11]

P. G. Kevrekidis, K. Ø. Rasmussen and A. R. Bishop, The discrete nonlinear Schrödinger equation: A survey of recent results, Int. J. Mod. Phys. B, 15 (2001), 2833-2900. doi: 10.1142/S0217979201007105.

[12]

A. Khare, K. Ø. Rasmussen, M. R. Samuelsen and A. Saxena, Exact solutions of the saturable discrete nonlinear Schrödinger equation, J. Phys. A, 38 (2005), 807-814. doi: 10.1088/0305-4470/38/4/002.

[13]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.

[14]

R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7 (1994), 1623-1643. doi: 10.1088/0951-7715/7/6/006.

[15]

A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 19 (2006), 27-40. doi: 10.1088/0951-7715/19/1/002.

[16]

A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach, Discret. Contin. Dyn. Syst., 19 (2007), 419-430. doi: 10.3934/dcds.2007.19.419.

[17]

A. Pankov and V. M. Rothos, Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 3219-3236. doi: 10.1098/rspa.2008.0255.

[18]

A. Pankov and N. Zakharchenko, On some discrete variational problems, Acta Appl. Math., 65 (2001), 295-303. doi: 10.1023/A:1010655000447.

[19]

J. A. Pava, "Nonlinear Dispersive Equations. Existence and Stability of Solitary and Periodic Travelling Wave Solutions," Mathematical Surveys and Monographs, 156, American Mathematical Society, Providence, RI, 2009.

[20]

D. E. Pelinovsky, P. G. Kevrekidis and D. J. Frantzeskakis, Stability of discrete solitons in nonlinear Schrödinger lattices, Phys. D, 212 (2005), 1-19. doi: 10.1016/j.physd.2005.07.021.

[21]

D. E. Pelinovsky and V. M. Rothos, Bifurcations of travelling wave solutions in the discrete NLS equations, Physica D, 202 (2005), 16-36. doi: 10.1016/j.physd.2005.01.016.

[22]

M. A. Porter, Experimental results related to DNLS equations, in "The Discrete Nonlinear Schrödinger Equation" (ed. P. G. Kevrekidis), 175-189, Springer Tracts Modern Phys., 232, Springer, Berlin, 2009.

[23]

W.-X. Qin and X. Xiao, Homoclinic orbits and localized solutions in nonlinear Schrödinger lattices, Nonlinearity, 20 (2007), 2305-2317. doi: 10.1088/0951-7715/20/10/002.

[24]

H. Shi and H. Zhang, Existence of gap solitons in periodic discrete nonlinear Schrödinger equations, J. Math. Anal. Appl., 361 (2010), 411-419. doi: 10.1016/j.jmaa.2009.07.026.

[25]

C. A. Stuart, Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation, Milan J. Math., 76 (2008), 329-399. doi: 10.1007/s00032-008-0089-9.

[26]

M. I. Weinstein, Excitation thresholds for nonlinear localized modes on lattices, Nonlinearity, 12 (1999), 673-691. doi: 10.1088/0951-7715/12/3/314.

[27]

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.

[28]

J. Yang and T. I. Lakoba, Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations, Stud. Appl. Math., 118 (2007), 153-197. doi: 10.1111/j.1467-9590.2007.00371.x.

[29]

G. Zhang and F. Liu, Existence of breather solutions of the DNLS equations with unbounded potentials, Nonlinear Anal.-Theory Methods Appl., 71 (2009), e786-e792. doi: 10.1016/j.na.2008.11.071.

[30]

G. Zhang and A. Pankov, Standing wave solutions of the discrete non-linear Schrödinger equations with unbounded potentials, to appear in Applicable Analysis, 2009.

show all references

References:
[1]

S. Aubry, Breathers in nonlinear lattices: Existence, linear stability and quantization, Lattice Dynamics (Paris, 1995), Physica D, 103 (1997), 201-250. doi: 10.1016/S0167-2789(96)00261-8.

[2]

W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697. doi: 10.1137/S1064827503422956.

[3]

J. Dorignac, J. Zhou and D. K. Campbell, Discrete breathers in nonlinear Schrödinger hypercubic lattices with arbitrary power nonlinearity, Physica D, 237 (2008), 486-504. doi: 10.1016/j.physd.2007.09.018.

[4]

J. Ch. Eilbeck and M. Johansson, The discrete nonlinear Schrödinger equation - 20 years on, in Proceedings of the Third Conference on Localization and Energy Transfer in Nonlinear Systems 2002, 44-87, 2003.

[5]

J. C. Eilbeck, P. S. Lomdahl and A. C. Scott, The discrete self-trapping equation, Physica D, 16 (1985), 318-338. doi: 10.1016/0167-2789(85)90012-0.

[6]

S. Flach, K. Kladko and R. S. MacKay, Energy thresholds for discrete breathers in one-, two-, and three-dimensional lattices, Phys. Rev. Lett., 78 (1987), 1207-1210. doi: 10.1103/PhysRevLett.78.1207.

[7]

M. Herrmann, Heteroclinic standing waves in defocussing DNLS equations: Variational approach via energy minimization, Appl. Anal., 89 (2010), 1591-1602, arXiv:1002.1591.

[8]

P. G. Kevrekidis, "The Discrete Nonlinear Schrödinger Equation," Springer Tracts in Modern Physics, 232, Chapter 1, General Introduction and Derivation of the DNLS Equation, 3-9, Springer, Berlin, 2009.

[9]

P. G. Kevrekidis, "The Discrete Nonlinear Schrödinger Equation," Springer Tracts in Modern Physics, 232, Chapter 5, The Defocusing Case, 117-141. Springer, Berlin, 2009.

[10]

P. G. Kevrekidis, "The Discrete Nonlinear Schrödinger Equation," Springer Tracts in Modern Physics, 232, Chapter 2, The One-Dimensional Case, 3-9, Springer, Berlin, 2009.

[11]

P. G. Kevrekidis, K. Ø. Rasmussen and A. R. Bishop, The discrete nonlinear Schrödinger equation: A survey of recent results, Int. J. Mod. Phys. B, 15 (2001), 2833-2900. doi: 10.1142/S0217979201007105.

[12]

A. Khare, K. Ø. Rasmussen, M. R. Samuelsen and A. Saxena, Exact solutions of the saturable discrete nonlinear Schrödinger equation, J. Phys. A, 38 (2005), 807-814. doi: 10.1088/0305-4470/38/4/002.

[13]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.

[14]

R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7 (1994), 1623-1643. doi: 10.1088/0951-7715/7/6/006.

[15]

A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 19 (2006), 27-40. doi: 10.1088/0951-7715/19/1/002.

[16]

A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach, Discret. Contin. Dyn. Syst., 19 (2007), 419-430. doi: 10.3934/dcds.2007.19.419.

[17]

A. Pankov and V. M. Rothos, Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 3219-3236. doi: 10.1098/rspa.2008.0255.

[18]

A. Pankov and N. Zakharchenko, On some discrete variational problems, Acta Appl. Math., 65 (2001), 295-303. doi: 10.1023/A:1010655000447.

[19]

J. A. Pava, "Nonlinear Dispersive Equations. Existence and Stability of Solitary and Periodic Travelling Wave Solutions," Mathematical Surveys and Monographs, 156, American Mathematical Society, Providence, RI, 2009.

[20]

D. E. Pelinovsky, P. G. Kevrekidis and D. J. Frantzeskakis, Stability of discrete solitons in nonlinear Schrödinger lattices, Phys. D, 212 (2005), 1-19. doi: 10.1016/j.physd.2005.07.021.

[21]

D. E. Pelinovsky and V. M. Rothos, Bifurcations of travelling wave solutions in the discrete NLS equations, Physica D, 202 (2005), 16-36. doi: 10.1016/j.physd.2005.01.016.

[22]

M. A. Porter, Experimental results related to DNLS equations, in "The Discrete Nonlinear Schrödinger Equation" (ed. P. G. Kevrekidis), 175-189, Springer Tracts Modern Phys., 232, Springer, Berlin, 2009.

[23]

W.-X. Qin and X. Xiao, Homoclinic orbits and localized solutions in nonlinear Schrödinger lattices, Nonlinearity, 20 (2007), 2305-2317. doi: 10.1088/0951-7715/20/10/002.

[24]

H. Shi and H. Zhang, Existence of gap solitons in periodic discrete nonlinear Schrödinger equations, J. Math. Anal. Appl., 361 (2010), 411-419. doi: 10.1016/j.jmaa.2009.07.026.

[25]

C. A. Stuart, Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation, Milan J. Math., 76 (2008), 329-399. doi: 10.1007/s00032-008-0089-9.

[26]

M. I. Weinstein, Excitation thresholds for nonlinear localized modes on lattices, Nonlinearity, 12 (1999), 673-691. doi: 10.1088/0951-7715/12/3/314.

[27]

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.

[28]

J. Yang and T. I. Lakoba, Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations, Stud. Appl. Math., 118 (2007), 153-197. doi: 10.1111/j.1467-9590.2007.00371.x.

[29]

G. Zhang and F. Liu, Existence of breather solutions of the DNLS equations with unbounded potentials, Nonlinear Anal.-Theory Methods Appl., 71 (2009), e786-e792. doi: 10.1016/j.na.2008.11.071.

[30]

G. Zhang and A. Pankov, Standing wave solutions of the discrete non-linear Schrödinger equations with unbounded potentials, to appear in Applicable Analysis, 2009.

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