Localized Structures in Dissipative Nonlinear Systems - Abstract
Malomed, Boris
A variety of the cubic complex Ginzburg-Landau (CGL) equations is considered,
featuring the background linear loss and a locally applied gain. The equations
of this type appertain to laser cavities based on planar waveguides, and to the description of thermal convection in binary fluids. With the gain localization
accounted for by a single delta-function (alias “hot spot”, HS), a solution for
pinned dissipative solitons is found in an exact analytical form, with one relation imposed
on parameters of the model [1]. The exponentially localized solution becomes weakly
(algebraically) localized in the limit case of the vanishing background loss.
Numerical solutions, with the delta-function replaced by a finite-width regularization,
demonstrate stability of the pinned solitons and their existence in the general case,
when the analytical solution is not available. If the gain-localization region and
the size of the soliton are comparable, the static soliton is replaced by a robust
breather [1]. A pair of two symmetric HSs is considered too. Numerical simulations
demonstrate that stable modes supported by the HS pair tend to be symmetric. An
unexpected conclusion is that the interaction between breathers pinned by two broad
HSs, which are the only stable modes in isolation in that case, transforms them into
a static symmetric mode [2]. Another case when exact solutions for pinned dissipative
solitons are available includes the Kerr nonlinearity concentrated at the HS, together
with the local gain and, possibly, with the nonlinear loss. Numerical tests demonstrate
that these pinned solitons obey a simple stability criterion if the localized nonlinear
loss is not included: they are stable/unstable if the localized nonlinearity is
self-defocusing/self-focusing [2].
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[1] C.-K. Lam, B.A. Malomed, K.W. Chow, and P.K.A. Wai, “Spatial solitons supported by localized gain in nonlinear optical waveguides“, Eur. Phys. J. Special Topics 173, 233-243 (2009).
relax [2] C.H. Tsang, B.A. Malomed, C.K. Lam, and K.W. Chow,“Solitons pinned to hot spot”,Eur. Phys. J. D (a special issue on “Dissipative Soliton”), DOI: 10.1140/epjd/e2010-00073-0.
Authors: C.-K. Lam, C.H. Tsang, and K. W. Chow (Department of Mechanical Engineering, The University of Hong Kong, Hong Kong), Boris A. Malomed (Department of Physical Electronics, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel)