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Exact solutions for periodic and solitary matter waves in nonlinear lattices
Dark solitary waves in nonlocal nonlinear Schrödinger systems
1. | Dpto. de Matemática Aplicada, Fac. CC. Químicas, Universidad Complutense de Madrid 28040, Spain |
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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