Dissipative Soliton - Theoretical Description of DSs

Theoretical Description of DSs

Most systems showing DSs are described by nonlinear partial differential equations. Discrete difference equations and cellular automata are also used. Up to now, modeling from first principles followed by a quantitative comparison of experiment and theory has been performed only rarely and sometimes also poses severe problems because of large discrepancies between microscopic and macroscopic time and space scales. Often simplified prototype models are investigated which reflect the essential physical processes in a larger class of experimental systems. Among these are

  • Reaction–diffusion systems, used for chemical systems, gas-discharges and semiconductors. The evolution of the state vector q(x, t) describing the concentration of the different reagents is determined by diffusion as well as local reactions:
\partial_t \boldsymbol{q} = \underline{\boldsymbol{D}} \Delta \boldsymbol{q} + \boldsymbol{R}(\boldsymbol{q}).
A frequently encountered example is the two-component Fitzhugh-Nagumo-type activator-inhibitor system
\left( \begin{array}{c} \tau_u \partial_t u\\\tau_v \partial_t v \end{array} \right) = \left(\begin{array}{cc} d_u^2 &0\\0&d_v^2 \end{array}\right) \left( \begin{array}{c} \Delta u\\ \Delta v \end{array} \right) + \left(\begin{array}{c} \lambda u -u^3 - \kappa_3 v +\kappa_1\\u-v \end{array}\right) .
Stationary DSs are generated by production of material in the center of the DSs, diffusive transport into the tails and depletion of material in the tails. A propagating pulse arises from production in the leading and depletion in the trailing end. Among other effects, one finds periodic oscillations of DSs ("breathing"), bound states, and collisions, merging, generation and annihilation.
  • Ginzburg-Landau type systems for a complex scalar q(x, t) used to describe nonlinear optical systems, plasmas, Bose-Einstein condensation, liquid crystals and granular media. A frequently found example is the cubic-quintic subcritical Ginzburg-Landau equation
\partial_t q = (d_r+ i d_i) \Delta q + l_r q + (c_r + i c_i)|q|^2 q + (q_r + i q_i) |q|^4 q.
To understand the mechanisms leading to the formation of DSs, one may consider the energy ρ = |q|2 for which one may derive the continuity equation
\partial_t \rho + \nabla \cdot \boldsymbol{m} = S = d_r (q \Delta q^{\ast} + q^{\ast} \Delta q) + 2 l_r \rho + 2 c_r \rho^2 + 2 q_r \rho^3 \quad\text{with} \quad\boldsymbol{m} = 2 d_i \text{Im}(q^{\ast}\nabla q).
One can thereby show that energy is generally produced in the flanks of the DSs and transported to the center and potentially to the tails where it is depleted. Dynamical phenomena include propagating DSs in 1d, propagating clusters in 2d, bound states and vortex solitons, as well as "exploding DSs".
  • The Swift-Hohenberg equation is used in nonlinear optics and in the granular media dynamics of flames or electroconvection. Swift-Hohenberg can be considered as an extension of the Ginzburg-Landau equation. It can be written as
\partial_t q = (s_r+ i s_i) \Delta^2 q + (d_r+ i d_i) \Delta q + l_r q + (c_r + i c_i)|q|^2 q + (q_r + i q_i) |q|^4 q.
For dr > 0 one essentially has the same mechanisms as in the Ginzburg-Landau equation. For dr < 0, in the real Swift-Hohenberg equation one finds bistability between homogeneous states and Turing patterns. DSs are stationary localized Turing domains on the homogeneous background. This also holds for the complex Swift-Hohenberg equations; however, propagating DSs as well as interaction phenomena are also possible, and observations include merging and interpenetration.
  • Space-time plots showing the dynamics and interaction of DSs as numerical solutions of the above model equations in one spatial dimension

  • Single "breathing" DS as solution of the two-component reaction-diffusion system with activator u (left half) and inhibitor v (right half).

  • Collision and merging of two DSs with a mutual phase difference of π/4 in the cubic-quintic Ginzburg-Landau equation, the plot shows the amplitude |q|.

  • "Interpenetration" of two DSs with a mutual phase difference of 0 in the Swift-Hohenberg equation with dr < 0, the plot shows the amplitude |q|.

Read more about this topic:  Dissipative Soliton

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