October  2011, 4(5): 1327-1340. doi: 10.3934/dcdss.2011.4.1327

Dark solitary waves in nonlocal nonlinear Schrödinger systems

1. 

Dpto. de Matemática Aplicada, Fac. CC. Químicas, Universidad Complutense de Madrid 28040, Spain

Received  September 2009 Revised  January 2010 Published  December 2010

Dark soliton-like solutions are analyzed in the context of a certain nonlocal nonlinear Schrödinger Equation with nonlocal dispersive term of Kac-Baker type. Main purpose is to investigate such solutions with negative nonlinear term and the presence of general integral dispersive terms. First the model is presented and the properties of the fundamental solution, the continuous wave, is studied. Dark solitary waves are perturbations of this plane wave. The study of dark type of solutions is divided in two different cases black and dark solitary waves. The range of existence of such solutions is studied analytically, and also their physical quantities like norm, momentum and energy. Usual behavior of nonlinear systems under nonlocal dispersive terms is found.
Citation: David Usero. Dark solitary waves in nonlocal nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1327-1340. doi: 10.3934/dcdss.2011.4.1327
References:
[1]

G. P. Agrawal, "Nonlinear Fiber Optics," Academic Press, Orlando, 1989.

[2]

G. L. Alfimov, V. M. Eleonsky and N. E. Kulagin, Dynamical systems in the theory of solitons in the presence of nonlocal interactions, Chaos, 2 (1992), 565-570. doi: 10.1063/1.165862.

[3]

G. L. Alfimov, V. M. Eleonsky, N. E. Kulagin and N. V. Mitskevich, Dynamics of topological solitons in models with nonlocal interactions, Chaos, 3 (1993), 405-414. doi: 10.1063/1.165948.

[4]

G. L. Alfimov, V. M. Eleonsky and L. Lerman, Solitary wave solutions of nonlocal sine-Gordon equations, Chaos, 8 (1998), 257-271. doi: 10.1063/1.166304.

[5]

G. L. Alfimov, T. Pierantozzi and L. Vázquez, Numerical study of a fractional sine-Gordon equation, in "Fractional Differentiation And its Applications" (eds. A. Le Mahaute, J. A. Tenreiro Machado, J. C. Trigeassou and J. Sabatier), Bordeaux, (2004),644-649.

[6]

G. L. Alfimov, D. Usero and L. Vázquez, On complex singularities of solutions of the equation $\mathcalHu_x-u+u^p=0$, J. Phys. A: Math Gen., 33 (2000), 6707-6720. doi: 10.1088/0305-4470/33/38/305.

[7]

G. A. Baker, One-dimensional order-disorder model wich approaches a second order phase transition, Phys. Rev., 122 (1961), 1477-1484. doi: 10.1103/PhysRev.122.1477.

[8]

I. V. Barashenkov, Stability criterion for dark solitons, Phys. Rev. Lett., 77 (1996), 1193-1197. doi: 10.1103/PhysRevLett.77.1193.

[9]

Ph. Blanchard, J. Stubbe and L. Vázquez, On the stability of solitary waves for classical scalar fields, Ann. Inst. Henri Poincaré, 47 (1987), 309-336.

[10]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. doi: 10.1017/S002211206700103X.

[11]

L. Di Menza and C. Gallo, The black solitons of the one-dimensional NLS equations, Nonlinearity, 20 (2007), 461-496. doi: 10.1088/0951-7715/20/2/010.

[12]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang and W. Krolikowski, Observation of attraction between dark solitons, Phys. Rev. Lett., 96 (2006), 043901. doi: 10.1103/PhysRevLett.96.043901.

[13]

Y. Gaididei, S. F. Mingaleev, P. L. Christiansen and K. O. Rasmussen, Effect of nonlocal dispersion on self interacting excitations, Phys. Lett. A, 222 (1995), 152-156. doi: 10.1016/0375-9601(96)00591-9.

[14]

Y. Gaididei, S. F. Mingaleev, P. L. Christiansen and K. O. Rasmussen, Effect of nonlocal dispersive interactions on self-interacting excitations, Phys. Rev. E, 55 (1997), 6141-6150. doi: 10.1103/PhysRevE.55.6141.

[15]

A. Hasegawa, "Optical Solitons in Fibers,", Springer-Verlag, (). 

[16]

A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion, Appl. Phys. Lett., 23 (1973), 142-144. doi: 10.1063/1.1654836.

[17]

A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion, Appl. Phys. Lett., 23 (1973), 171-173. doi: 10.1063/1.1654847.

[18]

M. Kac and E. Helfand, Study of several lattice systems with long-range forces, J. Math. Phys., 4 (1963), 1078-1088. doi: 10.1063/1.1704037.

[19]

Y. S. Kivshar, Dark solitons in nonlinear optics, I.E.E.E. J. Quantum Electron., 28 (1993), 250-264. doi: 10.1109/3.199266.

[20]

Y. S. Kivshar and W. Krölikovski, Lagrangian approach for dark solitons, Opt. Comm., 114 (1995), 353-362. doi: 10.1016/0030-4018(94)00644-A.

[21]

Y. S. Kivshar and B. Luther-Davies, Dark optical solitons: Physics and applications, Phys. Rep., 298 (1998), 81-197. doi: 10.1016/S0370-1573(97)00073-2.

[22]

Y. S. Kivshar and X. Yang, Perturbation-induced dynamics of dark-solitons, Phys. Rev. E, 49 (1994), 1657-1670. doi: 10.1103/PhysRevE.49.1657.

[23]

V. V. Konotop and V. E. Vekslerchik, Direct peerturbation theory for dark solitons, Phys. Rev. E, 49 (1994), 2397-2407. doi: 10.1103/PhysRevE.49.2397.

[24]

W. Królikowski and O. Bang, Solitons in nonlocal nonlinear media: Exact solutions, Phys. Rev. E, 63 (2000), 016610. doi: 10.1103/PhysRevE.63.016610.

[25]

A. G. Litvak, V. A. Mironov, G. M. Fraiman and A. D Yunakovskii, Thermal self-interaction of wave beams in a plasma with nonlocal nonlinearity, Sov. J. Plasma Phys., 1 (1975), 60-71.

[26]

S. F. Mingaleev, Y. B. Gaididei, E. Majernikova and S. Shpyrko, Kinks in the discrete sine-Gordon model with Kac-Baker long-range interactions, Phys. Rev. E, 61 (2000), 4454-4461. doi: 10.1103/PhysRevE.61.4454.

[27]

L. F. Mollenauer, R. H. Stolen and G. P. Gordon, Experimental observation of picosecond pulse narrowing and solitons in optical fibers, Phys. Rev. Lett., 45 (1980), 1095-1098. doi: 10.1103/PhysRevLett.45.1095.

[28]

H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. doi: 10.1143/JPSJ.39.1082.

[29]

A. Parola, L. Salasnich and L. Reatto, Structure and stability of bosonic clouds: Alkali-metal atoms with negative scattering length, Phys. Rev. A, 57 (1998), R3180.

[30]

D. E. Pelinovsky, Y. S. Kivshar and V. V. Afanasjev, Instability-induced dynamics of dark solitons, Phys. Rev. E, 54 (1996), 2015-2032. doi: 10.1103/PhysRevE.54.2015.

[31]

D. Suter and T. Blasberg, Stabilisation of transverse solitary waves by a nonlocal response of the nonlinear medium, Phys. Rev. A, 48 (1993), 4583-4587. doi: 10.1103/PhysRevA.48.4583.

[32]

D. Usero and L. Vázquez, Ecuaciones no locales y modelos fraccionarios, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, 99 (2005), 203-223.

[33]

N. G. Vakhitov and A. A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation, Radiophys. and Quantum Elect., 16 (1973), 783-789. doi: 10.1007/BF01031343.

[34]

L. Vázquez, W. A. B. Evans and G. Rickayzen, Numerical investigation of a non-local sine-Gordon model, Phys. Lett. A, 189 (1994), 454-459. doi: 10.1016/0375-9601(94)91209-2.

[35]

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird and W. J. Tomlinson, Experimental observation of the fundamental dark soliton in optical fibers, Phys. Rev. Lett., 61 (1988), 2445-2448. doi: 10.1103/PhysRevLett.61.2445.

[36]

G. B. Whitham, "Linear and Nonlinear Waves," Wiley Interscience, New York, 1974.

show all references

References:
[1]

G. P. Agrawal, "Nonlinear Fiber Optics," Academic Press, Orlando, 1989.

[2]

G. L. Alfimov, V. M. Eleonsky and N. E. Kulagin, Dynamical systems in the theory of solitons in the presence of nonlocal interactions, Chaos, 2 (1992), 565-570. doi: 10.1063/1.165862.

[3]

G. L. Alfimov, V. M. Eleonsky, N. E. Kulagin and N. V. Mitskevich, Dynamics of topological solitons in models with nonlocal interactions, Chaos, 3 (1993), 405-414. doi: 10.1063/1.165948.

[4]

G. L. Alfimov, V. M. Eleonsky and L. Lerman, Solitary wave solutions of nonlocal sine-Gordon equations, Chaos, 8 (1998), 257-271. doi: 10.1063/1.166304.

[5]

G. L. Alfimov, T. Pierantozzi and L. Vázquez, Numerical study of a fractional sine-Gordon equation, in "Fractional Differentiation And its Applications" (eds. A. Le Mahaute, J. A. Tenreiro Machado, J. C. Trigeassou and J. Sabatier), Bordeaux, (2004),644-649.

[6]

G. L. Alfimov, D. Usero and L. Vázquez, On complex singularities of solutions of the equation $\mathcalHu_x-u+u^p=0$, J. Phys. A: Math Gen., 33 (2000), 6707-6720. doi: 10.1088/0305-4470/33/38/305.

[7]

G. A. Baker, One-dimensional order-disorder model wich approaches a second order phase transition, Phys. Rev., 122 (1961), 1477-1484. doi: 10.1103/PhysRev.122.1477.

[8]

I. V. Barashenkov, Stability criterion for dark solitons, Phys. Rev. Lett., 77 (1996), 1193-1197. doi: 10.1103/PhysRevLett.77.1193.

[9]

Ph. Blanchard, J. Stubbe and L. Vázquez, On the stability of solitary waves for classical scalar fields, Ann. Inst. Henri Poincaré, 47 (1987), 309-336.

[10]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. doi: 10.1017/S002211206700103X.

[11]

L. Di Menza and C. Gallo, The black solitons of the one-dimensional NLS equations, Nonlinearity, 20 (2007), 461-496. doi: 10.1088/0951-7715/20/2/010.

[12]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang and W. Krolikowski, Observation of attraction between dark solitons, Phys. Rev. Lett., 96 (2006), 043901. doi: 10.1103/PhysRevLett.96.043901.

[13]

Y. Gaididei, S. F. Mingaleev, P. L. Christiansen and K. O. Rasmussen, Effect of nonlocal dispersion on self interacting excitations, Phys. Lett. A, 222 (1995), 152-156. doi: 10.1016/0375-9601(96)00591-9.

[14]

Y. Gaididei, S. F. Mingaleev, P. L. Christiansen and K. O. Rasmussen, Effect of nonlocal dispersive interactions on self-interacting excitations, Phys. Rev. E, 55 (1997), 6141-6150. doi: 10.1103/PhysRevE.55.6141.

[15]

A. Hasegawa, "Optical Solitons in Fibers,", Springer-Verlag, (). 

[16]

A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion, Appl. Phys. Lett., 23 (1973), 142-144. doi: 10.1063/1.1654836.

[17]

A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion, Appl. Phys. Lett., 23 (1973), 171-173. doi: 10.1063/1.1654847.

[18]

M. Kac and E. Helfand, Study of several lattice systems with long-range forces, J. Math. Phys., 4 (1963), 1078-1088. doi: 10.1063/1.1704037.

[19]

Y. S. Kivshar, Dark solitons in nonlinear optics, I.E.E.E. J. Quantum Electron., 28 (1993), 250-264. doi: 10.1109/3.199266.

[20]

Y. S. Kivshar and W. Krölikovski, Lagrangian approach for dark solitons, Opt. Comm., 114 (1995), 353-362. doi: 10.1016/0030-4018(94)00644-A.

[21]

Y. S. Kivshar and B. Luther-Davies, Dark optical solitons: Physics and applications, Phys. Rep., 298 (1998), 81-197. doi: 10.1016/S0370-1573(97)00073-2.

[22]

Y. S. Kivshar and X. Yang, Perturbation-induced dynamics of dark-solitons, Phys. Rev. E, 49 (1994), 1657-1670. doi: 10.1103/PhysRevE.49.1657.

[23]

V. V. Konotop and V. E. Vekslerchik, Direct peerturbation theory for dark solitons, Phys. Rev. E, 49 (1994), 2397-2407. doi: 10.1103/PhysRevE.49.2397.

[24]

W. Królikowski and O. Bang, Solitons in nonlocal nonlinear media: Exact solutions, Phys. Rev. E, 63 (2000), 016610. doi: 10.1103/PhysRevE.63.016610.

[25]

A. G. Litvak, V. A. Mironov, G. M. Fraiman and A. D Yunakovskii, Thermal self-interaction of wave beams in a plasma with nonlocal nonlinearity, Sov. J. Plasma Phys., 1 (1975), 60-71.

[26]

S. F. Mingaleev, Y. B. Gaididei, E. Majernikova and S. Shpyrko, Kinks in the discrete sine-Gordon model with Kac-Baker long-range interactions, Phys. Rev. E, 61 (2000), 4454-4461. doi: 10.1103/PhysRevE.61.4454.

[27]

L. F. Mollenauer, R. H. Stolen and G. P. Gordon, Experimental observation of picosecond pulse narrowing and solitons in optical fibers, Phys. Rev. Lett., 45 (1980), 1095-1098. doi: 10.1103/PhysRevLett.45.1095.

[28]

H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. doi: 10.1143/JPSJ.39.1082.

[29]

A. Parola, L. Salasnich and L. Reatto, Structure and stability of bosonic clouds: Alkali-metal atoms with negative scattering length, Phys. Rev. A, 57 (1998), R3180.

[30]

D. E. Pelinovsky, Y. S. Kivshar and V. V. Afanasjev, Instability-induced dynamics of dark solitons, Phys. Rev. E, 54 (1996), 2015-2032. doi: 10.1103/PhysRevE.54.2015.

[31]

D. Suter and T. Blasberg, Stabilisation of transverse solitary waves by a nonlocal response of the nonlinear medium, Phys. Rev. A, 48 (1993), 4583-4587. doi: 10.1103/PhysRevA.48.4583.

[32]

D. Usero and L. Vázquez, Ecuaciones no locales y modelos fraccionarios, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, 99 (2005), 203-223.

[33]

N. G. Vakhitov and A. A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation, Radiophys. and Quantum Elect., 16 (1973), 783-789. doi: 10.1007/BF01031343.

[34]

L. Vázquez, W. A. B. Evans and G. Rickayzen, Numerical investigation of a non-local sine-Gordon model, Phys. Lett. A, 189 (1994), 454-459. doi: 10.1016/0375-9601(94)91209-2.

[35]

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird and W. J. Tomlinson, Experimental observation of the fundamental dark soliton in optical fibers, Phys. Rev. Lett., 61 (1988), 2445-2448. doi: 10.1103/PhysRevLett.61.2445.

[36]

G. B. Whitham, "Linear and Nonlinear Waves," Wiley Interscience, New York, 1974.

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