GEOMETRIC PHASES AND MONODROMY AT SINGULARITIES (original) (raw)

On geometric phases for soliton equations

Communications in Mathematical Physics, 1992

This paper develops a new complex Hamiltonian structure for n-soliton solutions for a class of integrable equations such as the nonlinear Schrόdinger, sine-Gordon and Korteweg-de Vries hierarchies of equations that yields, amongst other things, geometric phases in the sense of Hannay and Berry. For example, one of the possible soliton geometric phases is manifested by the well known phase shift that occurs for interacting solitons. The main new tools are complex angle representations that linearize the corresponding Hamiltonian flows on associated noncompact Jacobi varieties. This new structure is obtained by taking appropriate limits of the differential equations describing the class of quasi-periodic solutions. A method of asymptotic reduction of the angle representations is introduced for investigating soliton geometric phases that are related to the presence of monodromy at singularities in the space of parameters. In particular, the phase shift of interacting solitons can be expressed as an integral over a cycle on an associated Riemann surface. In this setting, soliton geometric asymptotics are constructed for studying geometric phases in the quantum case. The general approach is worked out in detail for the three specific hierarchies of equations mentioned. Some links with τ-functions, the braid group and geometric quantization are pointed out as well.

Manifestation of Hamiltonian Monodromy in Nonlinear Wave Systems

Physical Review Letters, 2011

We show that the concept of dynamical monodromy plays a natural fundamental role in the spatiotemporal dynamics of counterpropagating nonlinear wave systems. By means of an adiabatic change of the boundary conditions imposed to the wave system, we show that Hamiltonian monodromy manifests itself through the spontaneous formation of a topological phase singularity (2-or-phase defect) in the nonlinear waves. This manifestation of dynamical Hamiltonian monodromy is illustrated by generic nonlinear wave models. In particular, we predict that its measurement can be realized in a direct way in the framework of a nonlinear optics experiment.

On the Geometry of Soliton Equations

Springer eBooks, 1995

The paper aims to suggest a geometric point of view in the theory of soliton equations. The belief is that a deeper understanding of the origin of these equations may provide a better understanding of their remarkable properties. According to the geometric point of view, soliton equations are the outcome of a specific reduction process of a bi-Hamiltonian manifold. The suggestion of the paper is to pay attention also to the 'unreduced form' of soliton equations.

The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

Letters in Mathematical Physics, 1994

The purpose of this letter is to investigate the geometry of new classes of solitonlike solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [1993] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and using special limiting procedures, draw some consequences from this setting. Among these consequences, one obtains new solutions such as quasiperiodic solutions, n-solitons, solitons with quasiperiodic background, billiard, and n-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow on N-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.

On soliton-type solutions of equations associated with N-component systems

Journal of Mathematical Physics, 2000

The algebraic geometric approach to N -component systems of nonlinear integrable PDE's is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to anti-kink transitions and multi-peaked soliton solutions is carried out. Transformations are used to connect these solutions to several other equations that model physical phenomena in fluid dynamics and nonlinear optics.

Matter-wave solitons with a periodic, piecewise-constant nonlinearity

2008

Motivated by recent proposals of ``collisionally inhomogeneous'' Bose-Einstein condensates (BECs), which have a spatially modulated scattering length, we study the existence and stability properties of bright and dark matter-wave solitons of a BEC characterized by a periodic, piecewise-constant scattering length. We use a ``stitching'' approach to analytically approximate the pertinent solutions of the underlying nonlinear Schr\"odinger equation by matching the wavefunction and its derivatives at the interfaces of the nonlinearity coefficient. To accurately quantify the stability of bright and dark solitons, we adapt general tools from the theory of perturbed Hamiltonian systems. We show that solitons can only exist at the centers of the constant regions of the piecewise-constant nonlinearity. We find both stable and unstable configurations for bright solitons and show that all dark solitons are unstable, with different instability mechanisms that depend on the ...

Bifurcations of phase portraits of a Singular Nonlinear Equation of the Second Class

Communications in Nonlinear Science and Numerical Simulation, 2014

The soliton dynamics is studied using the Frenkel Kontorova (FK) model with nonconvex interparticle interactions immersed in a parameterized on-site substrate potential. The case of a deformable substrate potential allows theoretical adaptation of the model to various physical situations. Non-convex interactions in lattice systems lead to a number of interesting phenomena that cannot be produced with linear coupling alone. In the continuum limit for such a model, the particles are governed by a Singular Nonlinear Equation of the Second Class. The dynamical behavior of traveling wave solutions is studied by using the theory of bifurcations of dynamical systems. Under different parametric situations, we give various sufficient conditions leading to the existence of propagating wave solutions or dislocation threshold, highlighting namely that the deformability of the substrate potential plays only a minor role.

Solitons supported by localized nonlinearities in periodic media

Physical Review A, 2011

Nonlinear periodic systems, such as photonic crystals and Bose-Einstein condensates (BECs) loaded into optical lattices, are often described by the nonlinear Schrödinger/Gross-Pitaevskii equation with a sinusoidal potential. Here, we consider a model based on such a periodic potential, with the nonlinearity (attractive or repulsive) concentrated either at a single point or at a symmetric set of two points, which are represented, respectively, by a single δ-function or a combination of two δ-functions. With the attractive or repulsive sign of the nonlinearity, this model gives rise to ordinary solitons or gap solitons (GSs), which reside, respectively, in the semi-infinite or finite gaps of the system's linear spectrum, being pinned to the δ-functions. Physical realizations of these systems are possible in optics and BEC, using diverse variants of the nonlinearity management. First, we demonstrate that the single δ-function multiplying the nonlinear term supports families of stable regular solitons in the self-attractive case, while a family of solitons supported by the attractive δ-function in the absence of the periodic potential is completely unstable. In addition, we show that the δ-function can support stable GSs in the first finite bandgap in both the self-attractive and repulsive models. The stability analysis for the GSs in the second finite bandgap is reported too, for both signs of the nonlinearity. Alongside the numerical analysis, analytical approximations are developed for the solitons in the semi-infinite and first two finite gaps, with the single δ-function positioned at a minimum or maximum of the periodic potential. In the model with the symmetric set of two δ-functions, we study the effect of the spontaneous symmetry breaking of the pinned solitons. Two configurations are considered, with the δ-functions set symmetrically with respect to the minimum or maximum of the underlying potential.