Simulations of the fusion of necklace-ring pattern in the complex Ginzburg-Landau equation by lattice Boltzmann method

Abstract A lattice Boltzmann model for solving the (2+1) dimensional cubic-quintic complex Ginzburg–Landau equation (CQCGLE) is proposed. Different from the classic lattice Boltzmann models, this lattice Boltzmann model is based on uniformly distributed lattice points in a two-dimensional space, and the evolution of the model is about a spatial axis rather than time. The algorithm provides advantages similar to the lattice Boltzmann method in that it is easily adapted to complex Ginzburg–Landau equations. Numerical results reproduce the phenomena of the fusion of necklace-ring pattern and the effect of non-linearity on the soliton in the CQCGLE.

[1]  I. Aranson,et al.  The world of the complex Ginzburg-Landau equation , 2001, cond-mat/0106115.

[2]  B. Malomed,et al.  Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation. , 2006, Physical review letters.

[3]  B. Malomed,et al.  Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses. , 2010, Physical review letters.

[4]  S Succi,et al.  Ground-state computation of Bose-Einstein condensates by an imaginary-time quantum lattice Boltzmann scheme. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  B. A. Malomed,et al.  Stable topological modes in two-dimensional Ginzburg-Landau models with trapping potentials , 2010, 1008.0224.

[6]  Guangwu Yan,et al.  A lattice Boltzmann model for the nonlinear Schrödinger equation , 2007 .

[7]  Y. Qian,et al.  Lattice BGK Models for Navier-Stokes Equation , 1992 .

[8]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases , 1939 .

[9]  Orazio Descalzi,et al.  Quasi-one-dimensional solutions and their interaction with two-dimensional dissipative solitons. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Guangwu Yan,et al.  Numerical Studies Based on Higher-Order Accuracy Lattice Boltzmann Model for the Complex Ginzburg-Landau Equation , 2012, J. Sci. Comput..

[11]  D. Wolf-Gladrow.  Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction , 2000 .

[12]  Cyrus K. Aidun,et al.  Lattice-Boltzmann Method for Complex Flows , 2010 .

[13]  Y J He,et al.  Self-trapped spatiotemporal necklace-ring solitons in the Ginzburg-Landau equation. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Irina Ginzburg,et al.  Variably saturated flow described with the anisotropic Lattice Boltzmann methods , 2006 .

[15]  B. Malomed,et al.  Stable vortex solitons supported by competing quadratic and cubic nonlinearities. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Boris A. Malomed,et al.  Spatiotemporal solitons in the Ginzburg-Landau model with a two-dimensional transverse grating , 2010 .

[17]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[18]  S Succi,et al.  Quantum lattice Boltzmann simulation of expanding Bose-Einstein condensates in random potentials. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Dhiraj V. Patil,et al.  Multigrid lattice Boltzmann method for accelerated solution of elliptic equations , 2014, J. Comput. Phys..

[20]  Boris A Malomed,et al.  Fusion of necklace-ring patterns into vortex and fundamental solitons in dissipative media. , 2007, Optics express.

[21]  Jeffrey Yepez,et al.  An efficient and accurate quantum lattice-gas model for the many-body Schrödinger wave equation , 2002 .

[22]  Segev,et al.  Self-trapping of "necklace-ring" beams in self-focusing kerr media , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  Sauro Succi.  Lattice Quantum Mechanics: An Application to Bose–Einstein Condensation , 1998 .

[24]  Guangwu Yan,et al.  Lattice Boltzmann model for the complex Ginzburg-Landau equation. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Boris A. Malomed,et al.  Solitary pulses in linearly coupled cubic quintic Ginzburg Landau equations , 2005 .

[26]  Jeffrey Yepez,et al.  Relativistic Path Integral as a Lattice-based Quantum Algorithm , 2005, Quantum Inf. Process..

[27]  Bastien Chopard,et al.  Lattice Boltzmann Computations and Applications to Physics , 1999, Theor. Comput. Sci..

[28]  D Mihalache,et al.  Robust soliton clusters in media with competing cubic and quintic nonlinearities. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[30]  K. M. Bryden,et al.  Parallel performance and accuracy of lattice Boltzmann and traditional finite difference methods for solving the unsteady two-dimensional Burger's equation , 2006 .

[31]  Akhmediev,et al.  Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[32]  Dumitru Mihalache,et al.  Generation of polygonal soliton clusters and fundamental solitons in dissipative systems by necklace-ring beams with radial-azimuthal phase modulation. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  Nail Akhmediev,et al.  Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion , 1997 .

[34]  Ping Dong,et al.  Lattice Boltzmann schemes for the nonlinear Schrödinger equation. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Marin Soljacic,et al.  Self-Trapping of ``Necklace'' Beams in Self-Focusing Kerr Media , 1998, physics/9810020.

[36]  Sauro Succi,et al.  Lattice Boltzmann equation for quantum mechanics , 1993, comp-gas/9304002.

[37]  He-Zhou Wang,et al.  Annularly and radially phase-modulated spatiotemporal necklace-ring patterns in the Ginzburg-Landau and Swift-Hohenberg equations. , 2009, Optics express.

[38]  V. Škarka,et al.  Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations. , 2006, Physical review letters.

[39]  Matthaeus,et al.  Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[40]  J. Boon.  The Lattice Boltzmann Equation for Fluid Dynamics and Beyond , 2003 .

[41]  Succi.  Numerical solution of the Schrödinger equation using discrete kinetic theory. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[42]  Guangwu Yan,et al.  Three-Dimensional Lattice Boltzmann Model for the Complex Ginzburg–Landau Equation , 2014, J. Sci. Comput..

[43]  Shiyi Chen,et al.  LATTICE BOLTZMANN METHOD FOR FLUID FLOWS , 2001 .