2005 Conference on Lasers and Electro-Optics Europe

Dissipative soliton interactions in laser systemsJ. M. Soto-Crespol and N. Akhmediev2

Instituto de 6ptica, C.S.L C., Serrano 121, 28006 Madrid, Spain2 Optical Sciences Group, RSPhysSE, ANU, ACT 0200, Australia

A bound state, formed by two dissipative solitons, may have a velocity which differs fromthat of a single soliton. As a result, these bound states can collide with single solitons with avariety of outcomes.

We present here numerical simulations of the collisions between a soliton pair and a single soliton in dissipativesystems described by the complex Ginzburg-Landau equation. We assign the numbers 1, 2 and 3 to the solitons,counting them from left to right (see Fig.la). Initially the solitons 1 and 2 form a stable pair. The soliton 3in the initial condition is well separated from the pair, so that there is no interaction with it. Nevertheless theresult of the interaction will depend on this initial distance and initial phase difference between the pair andthe single soliton. We can plot the separation between the solitons 2 and 3 and the phase difference betweenthem on the interaction plane in the same way as it is usually done for just two single solitons. This plot willrepresent the interaction plane of the initial conditions for the pair and a single soliton before the collision. Asummary of our results is presented in the Fig. lb. Initially, the point on the interaction plane for the solitons 2and 3 is located far away from the origin at a fixed distance (a: 6) in all our simulations. For the relative phasedifference, 0, we choose 36 values, with 10 degrees angular difference between neighbors. All 36 initial pointsare shown as small dots in Fig. lb. In this way, we cover a circle of initial conditions on the interaction plane ina relatively dense way. Each of these initial conditions leads to one of the outcomes that we shall describe. Thesimulations allowed us to establish a correspondence between the angular position of the initial point and theresult of the collision. Specifically, the points in the lower left quadrant result in the annihilation (or fusion)of one or two solitons. This is indicated in the figure close to each set of initial conditions. Fusion to a singlesoliton occurs when the initial point is in the upper part of this quadrant. Fusion of three solitons into twooccurs when the initial point is in the lower part of this quadrant.

7

(a)5-(bb

10 05 (-7lets5 0 3,5 7Solitonsclai

N ~ ~ ~ ~ ~ ~~~~~~~~~~~0 Fusion Trp

3->2

0 ~~~~~~-7-5 0 5 -7 -3.5 0 3.5 7

t P2-3coS( 2-3)FIG. 1. (a) Collision of two soliton bound state with a single soliton. We refer to this case as "soliton fusion".

Depending on the initial phase difference between the bound state and the single soliton, the output is one soliton.(b) Schematic of the evolution of the solitons 2 and 3 on the interaction plane in the three-soliton initial conditions. Theplot shows the outcomes of the collision for 36 initial conditions (small dots on a circle of radius z 6).

The rest of the circle of initial conditions can be divided into three arcs depending on the collision outcome.The final and initial points in this plane are joined schematically with straight lines. These are not the actualtrajectories. The trajectory may rotate around the origin before the collision. We obtain oscillating solitontriplets when the initial point is in the lower right quadrant of the interaction plane. As indicated in that figurethe final state in these cases is a limit cycle rather than a point. Moving soliton triplets are obtained when theinitial point is in the upper right quadrant. Finally, an "elastic collision" [1] occurs when the initial point is inthe upper part of the circle. If we select an initial distance between the solitons 2 and 3 which is different fromthe value 6, the whole circle of initial conditions may rotate around the origin thus moving our classificationscheme around. However, the relative location of the various outcomes of the collision will remain unchanged.

[1] Ph. Grelu and N. Akhmediev, Optics Express, 12, 3184 (2004)

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