Numerical continuation of invariant solutions of the complex Ginzburg–Landau equation

Abstract

We consider the problem of computation and deformation of group orbits of solutions of the complex Ginzburg-Landau equation (CGLE) with cubic nonlinearity in 1+1 space-time dimension invariant under the action of the three-dimensional Lie group of symmetries A(x, t) → e iθ A(x + σ, t + τ ). From an initial set of group orbits of invariant solutions, for a particular point in the parameter space of the CGLE, we obtain new sets of group orbits of invariant solutions via numerical continuation along paths in the moduli space. The computed solutions along the continuation paths are unstable, and have multiple modes and frequencies active in their spatial and temporal spectra, respectively. Structural changes in the moduli space resulting in symmetry gaining / breaking associated often with the spatial reflection symmetry A(x, t) → A(−x, t) of the CGLE were frequently uncovered in the parameter regions traversed.

Innovative solutions and sensitivity analysis of a fractional complex Ginzburg–Landau equation

“…Recently, the construction of dynamical systems and exact solutions, such as those for optical fiber communications, chemical reactions and biological systems, the shallow waters, plasmas and other fields can often be described by the nonlinear evolution equations [1,2,4,5,[9][10][11][12][13][14]. The Ginzburg-Landau (GL) equation is one of the most important NLPDEs in the field of mathematical physics, which was introduced into the study of superconductivity phenomenology theory by Ginzburg and Landau in the 20th century [15][16][17].…”

Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation

“…Turning to the GLE numerical solution, it should be stressed that, to the best of our knowledge, the conservation laws (integrals of motion) of this equation are absent in the literature (even for the linear case), despite extensive studies on the numerical methods having been conducted [63][64][65][66][67][68][69][70][71][72][73][74][75][76][77]. For developing difference schemes for the GLE solution, splitting methods (Strang splitting and alternating direction methods) were mostly used and the authors usually focused on the investigation of their convergence conditions and accuracy order.…”

“…The method seems to be an efficient tool for solving problems in domains of complicated geometry. Papers [67][68][69][70][71][72] were devoted to studying the stability of the GLE solution and its invariant solution for the chosen gauge and developing FDS for a numerical solution of the GLE involving the nonlocal nonlinear response of a medium. Paper [73] deals with the time-depended Landau-Khalatnikov equation modified in view of the Landau-Ginzburg-Devonshire approach.…”

New Exact Traveling Wave Solutions of the Time Fractional Complex Ginzburg-Landau Equation via the Conformable Fractional Derivative

“…The Ginzburg-Landau (GL) equation [1][2][3][4][5][6][7] is one of the most important partial differential equations in the field of mathematics and physics, which was introduced into the study of superconductivity phenomenology theory in the 20th century by Ginzburg and Landau. The GL equation usually describes the optical soliton [8][9][10][11][12][13][14][15] propagation through optical fibers over longer distances.…”