Bifurcation of asymmetric solutions in nonlinear optical media

Jones, Christopher K. R. T.; Küpper, Jones T.; Schaffner, Kenneth F.

doi:10.1007/pl00001578

Cited by 8 publications

(5 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…All of the examples of standing waves considered in [16] and [17] satisfy the hypotheses given here and thus enjoy a local decomposition of the flow by invariant manifolds. Some of these waveguide modes are stable, while others are unstable.…”

Section: Resultsmentioning

confidence: 83%

See 1 more Smart Citation

A spectral mapping theorem and invariant manifolds for nonlinear Schrodinger equations

Gesztesy¹,

Jones²,

Latushkin³

et al. 2000

Indiana Univ. Math. J.

View full text Add to dashboard Cite

Abstract. A spectral mapping theorem is proved that resolves a key problem in applying invariant manifold theorems to nonlinear Schrödinger type equations. The theorem is applied to the operator that arises as the linearization of the equation around a standing wave solution. We cast the problem in the context of space-dependent nonlinearities that arise in optical waveguide problems. The result is, however, more generally applicable including to equations in higher dimensions and even systems. The consequence is that stable, unstable, and center manifolds exist in the neighborhood of a (stable or unstable) standing wave, such as a waveguide mode, under simple and commonly verifiable spectral conditions. Main ResultsThe local behavior near some distinguished solution, such as a steady state, of an evolution equation, can be determined through a decomposition into invariant manifolds, that is, stable, unstable and center manifolds. These (locally invariant) manifolds are characterized by decay estimates. While the flows on the stable and unstable manifolds are determined by exponential decay in forward and backward time respectively, that on the center manifold is ambiguous. Nevertheless, a determination of the flow on the center manifold can lead to a complete characterization of the local flow and thus this decomposition, when possible, leads to a reduction of this problem to one of identifying the flow on the center manifold.This strategy has a long history for studying the local behavior near a critical point of an ordinary differential equation, or a fixed point of a map, and it has gained momentum in the last few decades in the context of nonlinear wave solutions of evolutionary partial differential equations. Extending the ideas to partial differential equations has, however, introduced a number of new issues. In infinite dimensions, the relation between the linearization and the full nonlinear equations is more delicate. This issue, however, turns out to be not so difficult for the invariant manifold decomposition and has largely been resolved, see, for instance, [2], [3]. A more subtle issue arises at the linear level. All of the known proofs for the existence of invariant manifolds are based upon the use of the group (or semigroup) generated by the linearization. The hypotheses of the relevant theorems are then formulated in terms of estimates on the appropriate projections of these groups onto stable, unstable and center subspaces. These amount to spectral estimates that come directly from a determination of the spectrum of the group. However, in any actual problem, the information available will, at best, be of the spectrum of the infinitesimal generator, that is, the linearized equation

show abstract

Section: Resultsmentioning

confidence: 83%

“…Using relation (16) for λ = ω 2 one concludes that (11) holds. The main tool in the second proof of Proposition 8 in the case n = 3 is Theorem I.23 in [24], inspired by previous results of Zemach and Klein [29].…”

Section: Proofsmentioning

confidence: 85%

A spectral mapping theorem and invariant manifolds for nonlinear Schrodinger equations

Gesztesy¹,

Jones²,

Latushkin³

et al. 2000

Indiana Univ. Math. J.

View full text Add to dashboard Cite

show abstract

“…Analogous equations appear in some mathematical models of nonlinear optics derived from Maxwell's equations [19]. For instance, in [20] the study of the propagation of electromagnetic waves in layered media leads to the scalar equation − + ( , 1 2 2 ) = 0, > 0,…”

Section: Introductionmentioning

confidence: 99%

“…For some other typical forms of ( , ), see [20] and the references therein. Also this class of equations has been widely investigated in the last decades [20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Complex Dynamics in One‐Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

Zanini

Zanolin

2018

Complexity

View full text Add to dashboard Cite

We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated.

show abstract

“…From a theoretical point of view, interesting properties, such as the induction of tunable spectroscopy as well as the generation of vortices and topological charges, can be studied. [4][5][6][7][8][9][10][11][12] In this paper, we present a study of the spatial evolution of optical fields generated by diffraction when a coordinate transformation is implemented in the transmittance function. This transformation generates nonlinear changes in the curvature function k 0 whose expression for a curve y ¼ fðxÞ is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 3 2 6 ; 4 5 7…”

Section: Introductionmentioning

confidence: 99%

Topological properties of optical fields: description of morphogenesis and bifurcations in focusing regions

et al. 2017

View full text Add to dashboard Cite

We analyze the diffraction field when changes in the curvature function of the boundary condition are implemented. The study is performed using differential geometry models with a curvature function displaying local behavior. Depending on the sign of curvature, we classify the diffraction field as elliptic, hyperbolic, or parabolic. In particular, it is shown that the optical field is organized around the parabolic regions, which correspond to focusing regions. The model is experimentally corroborated by applying a coordinate transformation to the transmittance of a zone plate. The reason to use this transmittance comes from the fact that its diffraction field displays multiple foci allowing identification, description, and control of bifurcations and morphogenesis effects, which are studied using the curvature function.

show abstract

Bifurcation of asymmetric solutions in nonlinear optical media

Cited by 8 publications

References 5 publications

A spectral mapping theorem and invariant manifolds for nonlinear Schrodinger equations

A spectral mapping theorem and invariant manifolds for nonlinear Schrodinger equations

Complex Dynamics in One‐Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

Topological properties of optical fields: description of morphogenesis and bifurcations in focusing regions

Contact Info

Product

Resources

About