Effect of Nonlinear Gradient Terms on Breathing Localized Solutions in the Quintic Complex Ginzburg-Landau Equation

Abstract

We study the effect of nonlinear gradient terms on breathing localized solutions in the complex Ginzburg-Landau equation. It is found that even small nonlinear gradient terms — which appear at the same order as the quintic term — can cause dramatic changes in the behavior of the solution, such as causing opposite sides of an otherwise monoperiodic symmetrically breathing solution to breathe at different frequencies, thus causing the solution to breathe periodically or chaotically on only one side or the solution to rapidly spread. [S0031-9007(98)07488-2] PACS numbers: 47.20.Ky, 03.40.Gc, 03.40.Kf, 05.70.Ln For over thirty years now it has been known that stable localized solutions can exist for certain nonlinear partial differential equations. The best-known example of such solutions is the soliton [1,2], a localized solution which occurs in purely dispersive systems such as the nonlinear Schrodinger equation. More recently stable localized solutions have been found to occur in quintic complex Ginzburg-Landau (CGL) equations [3‐ 7] —generic equations with both dissipation and dispersion and which describe systems near a subcritical bifurcation to traveling waves. These dissipative-dispersive localized (DDL) solutions can be considered to be the analog of the solitons that occur in purely dispersive systems. Although these DDL solutions share some properties with solitons, such as a fixed shape for the modulus and interaction behavior in which shape and size are preserved during collisions [4,5], there are fundamental differences. For example, these DDL solutions also exhibit mutual annihilation during collisions [4,5], a property which does not occur for solitons. Also, in contrast to solitons which require no energy input for their existence, the DDL solutions depend on a constant influx of energy in order to overcome the dissipation. Stable DDL solutions have also been studied in a two-dimensional (2D) quintic CGL equation [3,8], in a 2D equation for systems with broken rotational symmetry [9], and in equations describing systems in nonlinear optics — a dye laser with saturable absorber [10] and a system exhibiting optical bistability [11]. Experimentally, stable DDL solutions have been found in binary fluid convection [12,13] and in a dye laser with saturable absorber [14]. Until recently the behavior of localized solutions of prototype equations has been limited to solutions with fixed modulus such as the solitons and DDL solutions discussed above, or to solutions which oscillate periodically about zero (for real equations) such as the “breathers” of the sine Gordon equation. Therefore, an interesting discovery was that of stable localized solutions for which the modulus breathes periodically, quasiperiodically, or even chaotically [15]. By stable is meant that the solution lies on an attractor. These breathing DDL solutions, which were found for the quintic CGL equation, bare no relation to the “breathers” of the sine-Gordon equation. Also they are very different from the slowly spreading chaotic localized solutions of the quintic CGL equation [16,17]. The breathing DDL solutions exhibit interesting interaction behavior such as dependence on initial conditions for the outcome of collisions and even sensitive dependence on initial conditions for the outcome of collisions involving chaotic breathing DDL solutions [18]. For the breathing DDL solutions of the quintic CGL equation, nonlinear gradient terms have thus far been neglected for simplicity. However, for an actual physical system nonlinear gradient terms will always be present since they occur at the same order as the quintic term [19]. Therefore, an important question is whether there are any qualitative changes in the behavior as a result of the nonlinear gradient terms. In this Letter we study the effect of nonlinear gradient terms on the breathing DDL solutions of the quintic CGL equation. We find that even small nonlinear gradient terms can dramatically alter the behavior of solutions, such as causing opposite sides of an otherwise monoperiodic symmetrically breathing solution to breathe at different frequencies, causing the solution to breathe periodically or chaotically on only one side, or causing the solution to rapidly spread. We find that it is also possible for nonlinear gradient terms to cause an otherwise fixed-shape solution to breathe periodically or even chaotically. The quintic CGL equation with nonlinear gradient terms reads