Over the last few years it has become clear that the cubic-quintic complex Ginzburg-Landau (CGL) equation is becoming more and more important for the description of dissipative optical solitons. The cubic-quintic CGL equation is known to arise as a prototype envelope equation near the onset of a weakly inverted oscillatory. In 1991 Sakaguchi reported the stable existence of two classes of hole solutions for the cubic-quintic CGL equation. Quite recently we have shown that additional classes of hole solutions including breathing holes stably exist for the cubic-quintic CGL equation. Much less work than on single localized solutions has been done for their interaction. For coupled cubic-quintic CGL equations it has been shown that, as a function of the cross-coupling between counter-propagating waves, interacting pulse solutions can annihilate, interpenetrate, form a stationary bound state of two pulses or lead to a transition to the spatially homogeneous solution.

Figure caption:

The figure shows the time evolution of the interaction of two stationary pulse solutions resulting in two counter-propagating holes.

We shall be interested in the following research topics:

(a) Numerical study of the interaction of stationary counter-propagating pulses (1-D). Complete phase diagram of the interaction.

(b) Numerical study of the interaction of counter-propagating holes (1-D).

(c) Generalization to 2-D.

People involved:

The research will be done by O. Descalzi (numerical work) with the collaboration of H. Brand (Bayreuth, Germany).

References:

[1] O. Descalzi, M. Argentina and E. Tirapegui, Stationary Localized Solutions in the Subcritical Complex Ginzburg-Landau Equation, International Journal of Bifurcation and Chaos 12, 2459-2465 (2002).

[2] O. Descalzi, M. Argentina and E. Tirapegui, Saddle-Node Bifurcation: An Appearance Mechanism of Pulses in the Subcritical Complex Ginzburg-Landau Equation, Physical Review E 67, 015601 (R) (2003).

[3] O. Descalzi, On the Stability of Localized Structures in the Subcritical Complex Ginzburg-Landau Equation, Physica A 327, 23-28 (2003).

[4] O. Descalzi, An analytical approach to nucleation solutions and pulses in the one- dimensional real and complex Ginzburg-Landau equations, in Instabilities and Non-equilibrium Structures IX, Eds. O. Descalzi, S. Rica and J. Martínez, Kluwer Academic Publishers, Dordrecht (2004), 149-184.

[5] O. Descalzi and E. Tirapegui, On the moving pulse solutions in systems with broken parity, Physica A 342, 9-15 (2004).

[6] O. Descalzi, G. Düring, and E. Tirapegui, On the stable stationary hole solutions in the complex Ginzburg-Landau equation, Physica A 356, 66-71 (2005).

[7] O. Descalzi, P. Gutiérrez and E. Tirapegui, Localized structures in nonequilibrium systems, International Journal of Modern Physics C 16 No. 12 1909-1916 (2005).

[8] O. Descalzi, Static, oscillating and moving pulses in the one-dimensional Ginzburg-Landau equation: an analytical approach, Physical Review E 72, 046210 (2005).

[9] O. Descalzi and H.R. Brand, Stable stationary and breathing holes at the onset of a weakly inverted instability, Physical Review E 72, 055202(R) (2005).