Multisoliton solutions and energy sharing collisions in coupled nonlinear Schrödinger equations with focusing, defocusing and mixed type nonlinearities

Bright and bright-dark type multisoliton solutions of the integrable N-coupled nonlinear Schrödinger (CNLS) equations with focusing, defocusing and mixed type nonlinearities are obtained by using Hirota’s bilinearization method. Particularly, for the bright soliton case, we present the Gram type determinant form of the n-soliton solution (n:arbitrary) for both focusing and mixed type nonlinearities and explicitly prove that the determinant form indeed satisfies the corresponding bilinear equations. Based on this, we also write down the multisoliton form for the mixed (bright-dark) type solitons. For the focusing and mixed type nonlinearities with vanishing boundary conditions the pure bright solitons exhibit different kinds of nontrivial shape changing/energy sharing collisions characterized by intensity redistribution, amplitude dependent phase-shift and change in relative separation distances. Due to nonvanishing boundary conditions the mixed N-CNLS system can admit coupled bright-dark solitons. Here we show that the bright solitons exhibit nontrivial energy sharing collision only if they are spread up in two or more components, while the dark solitons appearing in the remaining components undergo mere standard elastic collisions. Energy sharing collisions lead to exciting applications such as collision based optical computing and soliton amplification. Finally, we briefly discuss the energy sharing collision properties of the solitons of the (2+1) dimensional long wave-short wave resonance interaction (LSRI) system.

[1]  Gino Biondini,et al.  Optical solitons: Perspectives and applications. , 2000, Chaos.

[2]  Oktay K. Pashaev,et al.  On the integrability and isotopic structure of the one-dimensional Hubbard model in the long wave approximation , 1981 .

[3]  M Lakshmanan,et al.  Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  M. Lakshmanan,et al.  Bright and dark soliton solutions to coupled nonlinear Schrodinger equations , 1995 .

[5]  M Lakshmanan,et al.  Exact soliton solutions of coupled nonlinear Schrödinger equations: shape-changing collisions, logic gates, and partially coherent solitons. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Yuri S. Kivshar,et al.  Polarized dark solitons in isotropic Kerr media , 1997 .

[7]  Nick Lazarides,et al.  Coupled nonlinear Schrodinger field equations for electromagnetic wave propagation in nonlinear left-handed materials. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Alwyn C. Scott,et al.  Launching a Davydov Soliton: I. Soliton Analysis , 1984 .

[9]  Symbiotic solitons in heteronuclear multicomponent Bose-Einstein condensates , 2005, cond-mat/0506405.

[10]  Camille-Sophie Brès,et al.  Observation of temporal vector soliton propagation and collision in birefringent fiber. , 2007, Physical review letters.

[11]  Kenneth Steiglitz,et al.  State transformations of colliding optical solitons and possible application to computation in bulk media , 1998 .

[12]  Yuri S. Kivshar,et al.  Solitons in photonic crystals , 2003 .

[13]  Yuri S. Kivshar,et al.  Optical Solitons: From Fibers to Photonic Crystals , 2003 .

[14]  Kang,et al.  Observation of Manakov spatial solitons in AlGaAs planar waveguides. , 1996, Physical review letters.

[15]  M. Lakshmanan,et al.  Bright-dark solitons and their collisions in mixed N -coupled nonlinear Schrödinger equations , 2007, 0711.4424.

[16]  K. Steiglitz,et al.  Energy-Exchange Interactions between Colliding Vector Solitons , 1999 .

[17]  W. Ketterle,et al.  Observation of Feshbach resonances between two different atomic species. , 2004, Physical review letters.

[18]  A. Hasegawa.  An historical review of application of optical solitons for high speed communications. , 2000, Chaos.

[19]  V. K. Fedyanin,et al.  Solitons in a One-Dimensional Modified Hubbard Model† , 1978 .

[20]  K Steiglitz,et al.  Time-gated Manakov spatial solitons are computationally universal. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Adrian Ankiewicz,et al.  Solitons : nonlinear pulses and beams , 1997 .

[22]  B. Apagyi,et al.  Stability of static solitonic excitations of two-component Bose-Einstein condensates in finite range of interspecies scattering length a~1~2 (8 pages) , 2004 .

[23]  T. Hänsch,et al.  Sympathetic cooling of 85 Rb and 87 Rb , 2001 .

[24]  N-Soliton Collision in the Manakov Model , 2003, nlin/0302059.

[25]  Mark J. Ablowitz,et al.  On discretizations of the vector nonlinear Schrödinger equation , 1998, solv-int/9810014.

[26]  Miki Wadati,et al.  Exact analysis of soliton dynamics in spinor Bose-Einstein condensates. , 2004, Physical review letters.

[27]  M. Lakshmanan,et al.  Shape changing collisions of optical solitons, universal logic gates and partially coherent solitons in coupled nonlinear Schrödinger equations , 2001 .

[28]  Q. Park,et al.  Systematic construction of multicomponent optical solitons , 2000 .

[29]  V. Makhankov.  Quasi-classical solitons in the Lindner-Fedyanin model - “hole”-like excitations , 1981 .

[30]  M. Lakshmanan,et al.  Higher dimensional bright solitons and their collisions in a multicomponent long wave–short wave system , 2008, 0810.2868.

[31]  H. Herrero,et al.  Bose-Einstein solitons in highly asymmetric traps , 1998 .

[32]  M. Marklund,et al.  Solitons and decoherence in left-handed metamaterials , 2005 .

[33]  Q. Park,et al.  Systematic construction of vector solitons , 2002 .

[34]  M Lakshmanan,et al.  Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations. , 2001, Physical review letters.

[35]  Yasuhiro Ohta,et al.  Localized solitons of a (2 + 1)-dimensional nonlocal nonlinear Schrödinger equation , 2008, 0804.1406.

[36]  Jarmo Hietarinta,et al.  Inelastic Collision and Switching of Coupled Bright Solitons in Optical Fibers , 1997, solv-int/9703008.