Spinning solitons in cubic-quintic nonlinear media

Crasovan, L.-C.; Malomed, Boris A.; Mihalache, Dumitru

doi:10.1007/s12043-001-0013-0

Cited by 44 publications

(50 citation statements)

References 73 publications

(77 reference statements)

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“…Accordingly, the symmetric solutions emerge at k = λ, with the known minimum (threshold) values of the energy in the single component [16]: E thr = 11.73, 48.38, and 88.34, for s = 0, 1, and 2, respectively. Within the family of the symmetric solitons, k varies from k min ≡ λ, which corresponds to E = E thr , up to k max = λ + 3/16, corresponding to E → ∞ (k = 3/16 is the value of the propagation constant in the single-component 2D model at which the energy diverges, along with the soliton's radius, for any s [15]).…”

Section: Asymmetric Solitons and Bifurcation Loopsmentioning

confidence: 95%

“…In the experiment, it may be coupled into the planar waveguide by an oblique vortical laser beam shone onto the waveguide under an appropriate angle. In that sense, the physical purport of the spatiotemporal vortices is essentially different from that of (2 + 1)-dimensional spatial solitons with the embedded vorticity, which are understood as hollow cylindrical beams of light propagating in a bulk medium [11][12][13][14][15][16]. In the experiment, vortical spatial solitons, built as multi-beam complexes with the phase distribution carrying the effective vorticity (rather than cylindrical beams), were created in photorefractive crystals with the saturable nonlinearity, their stability against splitting being maintained by a photoinduced lattice potential [24]).…”

Section: The Modelmentioning

confidence: 99%

“…Quiroga-Teixeiro and Michinel [14] were the first to demonstrate in direct simulations that 2D solitons with s = 1 may be stable in the CQ model, provided that their power (norm) is sufficiently large. The detailed analysis [15,16], which made use of the linearized version of the model for computing growth rates of perturbation eigenmodes, had demonstrated that, for spins s = 1 and 2, the CQ model exhibits relatively large stability regions, which cover, respectively, ≈ 9% and ≈ 8% of the respective existence regions, in terms of soliton's norm (total power). It has also been demonstrated that the spinning solitons with the vorticity up to s = 5 may also be stable, but in extremely narrow regions [17].…”

Section: Introductionmentioning

confidence: 99%

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Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity

Dror

Malomed

2011

Physica D: Nonlinear Phenomena

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It is well known that the two-dimensional (2D) nonlinear Schrödinger equation (NLSE) with the cubic-quintic (CQ) nonlinearity supports a family of stable fundamental solitons, as well as solitary vortices (alias vortex rings), which are stable for sufficiently large values of the norm. We study stationary localized modes in a symmetric linearly coupled system of two such equations, focusing on asymmetric states. The model may describe "optical bullets" in dual-core nonlinear optical waveguides (including spatiotemporal vortices that were not discussed before), or a BoseEinstein condensate (BEC) loaded into a "dual-pancake" trap. Each family of solutions in the single-component model has two different counterparts in the coupled system, one symmetric and one asymmetric. Similarly to the earlier studied coupled 1D system with the CQ nonlinearity, the present model features bifurcation loops, for fundamental and vortex solitons alike: with the increase of the total energy (norm), the symmetric solitons become unstable at a point of the direct bifurcation, which is followed, at larger values of the energy, by the reverse bifurcation restabilizing the symmetric solitons. However, on the contrary to the 1D system, both the direct and reverse bifurcation may be of the subcritical type, at sufficiently small values of the coupling constant, λ. Thus, the system demonstrates a double bistability for the fundamental solitons. The stability of the solitons is investigated via the computation of instability growth rates for small perturbations. Vortex rings, which we study for two values of the "spin", s = 1 and 2, may be subject to the azimuthal instability, like in the single-component model. In particular, complete destabilization of asymmetric vortices is demonstrated for a sufficiently strong linear coupling. With the decrease of λ, a region of stable asymmetric vortices appears, and a single region of bistability for the vortices is found. We also develop a quasi-analytical approach to the description of the bifurcations diagrams, based on the variational approximation. Splitting of asymmetric vortices, induced by the azimuthal instability, is studied by means of direct simulations. Interactions between initially quiescent solitons of different types are studied too. In particular, we confirm the prediction of the reversal of the sign of the interaction (attractive/repulsive for in-phase/out-of-phase pairs) for the solitons with the odd spin, s = 1, in comparison with the even values, s = 0 and 2.

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Section: Asymmetric Solitons and Bifurcation Loopsmentioning

confidence: 95%

Section: The Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity

Dror

Malomed

2011

Physica D: Nonlinear Phenomena

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“…These occur at parameter values β = δ = 1 2 , = 2.5, μ = 1, ν = 0.1; see [7], [8]. The symmetry group is again the four-dimensional group SE(2) × S 1 ; cf.…”

Section: Quinticmentioning

confidence: 99%

“…In section 3 we will discuss several two-dimensional systems from the literature (e.g., Barkley's spiral system [1], [3], the (λ − ω)-system [18], and the quintic Ginzburg-Landau equation [7], [8]) that show rigidly rotating spiral waves. Freezing such waves can be delicate because it depends on the precise choice of phase condition (with or without weighted L 2 -norms), the type of numerical discretization (rectangular or polar grid), and on the right choice of the underlying group.…”

Section: 3) U(t) = A(γ(t))v(t)mentioning

confidence: 99%

Freezing Solutions of Equivariant Evolution Equations

Beyn¹,

Thümmler²

2004

SIAM J. Appl. Dyn. Syst.

155

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Abstract. In this paper we develop numerical methods for integrating general evolution equations ut = F (u), u(0) = u0, where F is defined on a dense subspace of some Banach space (generally infinitedimensional) and is equivariant with respect to the action of a finite-dimensional (not necessarily compact) Lie group. Such equations typically arise from autonomous PDEs on unbounded domains that are invariant under the action of the Euclidean group or one of its subgroups. In our approach we write the solution u(t) as a composition of the action of a time-dependent group element with a "frozen solution" in the given Banach space. We keep the frozen solution as constant as possible by introducing a set of algebraic constraints (phase conditions), the number of which is given by the dimension of the Lie group. The resulting PDAE (partial differential algebraic equation) is then solved by combining classical numerical methods, such as restriction to a bounded domain with asymptotic boundary conditions, half-explicit Euler methods in time, and finite differences in space.We provide applications to reaction-diffusion systems that have traveling wave or spiral solutions in one and two space dimensions.

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Dynamics of Patterns in Nonlinear Equivariant PDEs

Beyn

Thümmler

2009

GAMM-Mitteilungen

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Many solutions of nonlinear time dependent partial differential equations show particular spatio-temporal patterns, such as traveling waves in one space dimension or spiral and scroll waves in higher space dimensions. The purpose of this paper is to review some recent progress on the analytical and the numerical treatment of such patterns. Particular emphasis is put on symmetries and on the dynamical systems viewpoint that goes beyond existence, uniqueness and numerical simulation of solutions for single initial value problems. The nonlinear asymptotic stability of dynamic patterns is discussed and a numerical approach (the freezing method) is presented that allows to compute co-moving frames in which solutions converging to the patterns become stationary. The results are related to the theory of relative equilibria for equivariant evolution equations. We discuss several applications to parabolic systems with nonlinearities of FitzHugh-Nagumo and Ginzburg-Landau type.

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Spinning solitons in cubic-quintic nonlinear media

Abstract: We review recent theoretical results concerning the existence, stability and unique features of families of bright vortex solitons (doughnuts, or 'spinning' solitons) in both conservative and dissipative cubic-quintic nonlinear media.

Cited by 44 publications

References 73 publications

Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity

Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity

Freezing Solutions of Equivariant Evolution Equations

Dynamics of Patterns in Nonlinear Equivariant PDEs

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