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Publications [2002 ] [2001 ] [2000] [1999]
4. Continuum Mechanics and Hamiltonian PDEs
4.5. Stability and bifurcations of optical pulses
2002
Authors: P. Ashwin, M.V. Bartuccelli, T.J. Bridges & S.A. Gourley
Title: Travelling fronts for the KPP equation with spatio-temporal delay
Journal: Z. angew. Mathematik & Physik 53: 103-122
YEAR: 2002
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Authors: T.J. Bridges
Title: Stability of solitary waves: geometry, symplecticity and three-dimensionality
Journal: Benjamin Memorial Lecture, in Proceedings of 2001 IMA WOW Conference
Year: 2002
Authors: T.J. Bridges & G. Derks
Title: Dimension breaking of gradient elliptic operators
Journal: Conference on Nonlinear Analysis 2002 (in honour of K. Kirchgaessner's 70th birthday)
Year: 2002
Authors: K.B. Blyuss, T.J. Bridges & G. Derks
Title: Transverse instability and its long-term development for solitary waves of the (2+1)-Boussinesq equation.
Journal: Preprint
Year: 2002
Author(s): Thomas J. Bridges, Gianne Derks and Georg Gottwald
Title: Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework
Journal: To appear in Physica D.
YEAR: 2002
Abstract: The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation
is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a robust shooting algorithm on exterior algebra spaces is introduced. The complete algorithm has several novel features, including a rigorous algorithm for choosing starting values, the role of Grassmannian submanifolds in choosing the
numerical integrator, and the use of the Hodge star operator for deducing a numerically computable form for the Evans function. The algorithm is illustrated by computing the stability and instability of solitary waves of the fifth-order KdV equation with polynomial nonlinearity.
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Author(s): T.J. Bridges and G. Derks
Title: Linear instability of solitary wave solutions of the Kawahara equation and its generalizations
Journal: SIAM J. Math. Anal. 33, pp 1356-1378
YEAR: 2002
Abstract: The linear stability problem for solitary-wave states of the Kawahara - or fifth-order KdV-type - equation and its generalizations is considered. A new formulation of the stability problem in terms of the symplectic Evans matrix is presented. The formulation is based on a multi-symplectification of the Kawahara equation, and leads to a new characterization of the basic solitary wave, including changes in the state
at infinity represented by embedding the solitary wave in a multi-parameter family. The theory is used to give a rigorous geometric sufficient condition for instability. The theory is abstract and applies to a wide range of solitary-wave states. For example, the theory is applied to the families of solitary waves found by Kichenassamy-Olver and Levandosky.
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Author(s): T.J. Bridges and G. Derks
Title: Constructing the symplectic Evans Matrix using maximally analytic individual vectors
Journal: To appear in Roy. Soc. Edinburgh Proc. A .
YEAR: 2002
Abstract: For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian PDEs about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. However, in general this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity.
In this paper, a new construction of the symplectic Evans matrix is presented which is based on individual vectors but is
analytic at the branch points -- indeed maximally analytic. In fact this result has greater generality than just the symplectic case: it solves
the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when
individual vectors are used? Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors.
This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.
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Author(s): G. Gottwald, R. Grimshaw and B. Malomed
Title: Cuspons, peakons and regular gap solitons between three dispersion curves.
Journal: Phys. Rev. E (to appear)
YEAR: 2002
Abstract: A general model is introduced to describe a wave-envelope system for the situation when the linear dispersion relation has three branches, which in the absence of any coupling terms between these branches, would intersect pair-wise in three nearly-coincident points. The system contains two waves with a strong linear coupling between them, to which a third wave is
then coupled. This model has two gaps in its linear spectrum. As is typical for wave-envelope systems, the model also contains a set of cubic nonlinear terms. Realizations of this model can be made in terms of temporal or spatial evolution of optical fields in, respectively, either a planar waveguide, or a bulk-layered medium resembling a photonic-crystal fiber, which carry a triple spatial Bragg grating. Another physical system described by the same general model is a set of three internal wave modes in a density-stratified fluid, whose phase speeds come into close coincidence for a certain wavenumber. A nonlinear analysis is performed for zero-velocity solitons, that is, they have zero velocity in the reference frame in which the third wave has zero group velocity. If one may disregard the self-phase modulation (SPM) term in the equation for the third wave, we find an analytical solution which shows that there simultaneously exist two different families of solitons: regular ones, which may be regarded as a smooth deformation of the usual gap solitons in a two-wave system, and cuspons, which have finite amplitude and energy, but a singularity in the
first derivative at their center. Even in the limit when the linear coupling of the third wave to the first two nearly vanishes, the soliton family remains drastically different from that in the uncoupled system; in this limit, regular solitons whose amplitude exceeds a certain critical value are replaced by peakons. While the regular solitons, cuspons, and peakons are found in an exact analytical form, their stability is tested numerically, which shows that they all may be stable. If the SPM terms are retained, we find that there may again simultaneously exist two different families of generic stable soliton solutions, namely, regular ones and peakons. Direct simulations show that both types of solitons are stable in this case.
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Author(s): G. Gottwald, R. Grimshaw and B. Malomed
Title: Cuspons and peakons vis-a-vis regular solitons and collapse in a three-wave system.
Journal: Proceedings of the AMS-IMS-SIAM Conference "The Legacy of Inverse Scattering Theory in Nonlinear Wave Propagation CONM (Contemporary Math) AMS series(to appear)
YEAR: 2002
(MASIE)Subsection : 4.5
Abstract: We introduce a general model of a one-dimensional three-component wave system with cubic nonlinearity. Linear couplings between the components prevent intersections between the corresponding dispersion curves, which opens two gaps in the system's linear spectrum. Detailed analysis is performed for zero-velocity solitons, in the reference frame in which the group velocity of one wave is zero. Disregarding the self-phase-modulation (SPM) term in the equation for that wave, we find an analytical solution which shows that there simultaneously exist two different families of generic solitons: regular ones, which may be regarded as a smooth deformation of the usual gap solitons in the two-wave system, and cuspons with a singularity in the first derivative at the center, while their energy is finite. Even in the limit when the linear coupling of the zero-group-velocity wave to the other two components is vanishing, the soliton family remains drastically different from that in the linearly uncoupled system: in this limit, regular solitons whose amplitude exceeds a certain critical value are replaced by peakons. While the regular solitons, cuspons, and peakons are found in an exact analytical form, their stability is tested numerically, showing that they all may be stable. In the case when the cuspons are unstable, the instability may trigger onset of spatio-temporal collapse in the system. If the SPM terms are
retained, we find that there again simultaneously exist two different families of generic stable soliton solutions, which are regular ones and peakons. The existence of the peakons depends, in this case, on the sign of certain parameters of the system. Direct simulations show that both types of the solitons may be stable in this most general case too.
Author(s): Grimshaw, R., Malomed, B. and Gottwald, G.
Title: Singular and regular gap solitons between three dispersion curves.
Journal: Phys. Rev. E (to appear)
YEAR: 2002
Author(s): Grimshaw, R., Green, G.A. and Malomed, B.A.
Title: Cuspons and peakons vis-a-Vis regular solitons and collapse in a three-wave system.
Journal: Contempory Mathematics, ed. R. Choudhury (to appear)
YEAR: 2002
2001
Authors: T.J. Bridges
Title: Transverse instability of solitary-wave states of the water-wave problem
Journal: J. Fluid Mechanics 439: 255-278
YEAR: 2001
Authors: T.J. Bridges & F.E. Laine-Pearson
Title: Multi-symplectic relative equilibria, multi-phase wavetrains and coupled NLS equations,
Journal: Studies in Applied Mathematics 107: 137-155
Year: 2001
Author(s): T.J. Bridges and G. Derks
Title: The symplectic Evans matrix, and the instability of solitary waves and fronts with symmetry.
Journal: Arch. Rat. Mech. Anal. 156, pp 1-87, 2001
YEAR: 2001
Abstract: Hamiltonian evolution equations which are equivariant with respect to the action of a Lie group are models for physical phenomena such as oceanographic flows, optical fibres and atmospheric flows, and such systems often have a wide variety of solitary wave or front solutions. In this paper, we present a new symplectic framework for analyzing the spectral problem associated with the linearization about such solitary waves and fronts. At the heart of the analysis is a multi-symplectic formulation of Hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions, with a third symplectic structure - with two-form denoted by $\Omega$ - associated with a coordinate frame moving at the speed of the wave. This leads to a geometric decomposition and symplectification of the Evans function formulation for the linearization about solitary waves and fronts.
We introduce the concept of the symplectic Evans matrix, a matrix consisting of restricted $\Omega$-symplectic forms. By applying Hodge duality to the exterior algebra formulation of the Evans function, we find that the zeros of the Evans function correspond to zeros of the determinant of the symplectic Evans matrix. Based on this formulation, we prove several new properties of the Evans function. Restricting the spectral parameter $\lambda$ to the real axis, we obtain rigorous results on the derivatives of the Evans function near
the origin, based solely on the abstract geometry of the equations, and results for the large $|\lambda|$ behaviour which use primarily the symplectic structure, but also extend to the non-symplectic case.
The Lie group symmetry affects the Evans function by generating zero eigenvalues of large multiplicity in the so-called systems at infinity. We present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation.
By combining all these results, a new rigorous sufficient condition for instability of solitary waves and fronts is obtained.
The theory applies to a large class of solitary waves and fronts including waves which are biasymptotic to a nonconstant manifold of states as $|x|$ tends to infinity. To illustrate the theory, it is applied to three examples: a Boussinesq model from oceanography, a class of nonlinear Schrodinger equations from optics and a nonlinear Klein-Gordon equation from atmospheric dynamics.
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Author(s): T.J. Bridges and G. Derks
Title: Dimension breaking of nonlinear elliptic PDEs: satisfying the spectral condition geometrically
Preprint: UNIS preprint
YEAR: 2001
Abstract: Dimension breaking occurs when the solution of a nonlinear partial differential equation (PDE) depending on $n$ independent variables bifurcates to one depending on $n+1$. A central hypothesis in the theory of dimension breaking is that a certain operator should have a non-zero purely imaginary eigenvalue. This hypothesis is difficult to verify in general. We present a geometric theory for verifying this hypothesis. Moreover, for a large class of partial differential equations, namely multi-symplectic Hamiltonian PDEs,
we show that the verification of this hypothesis is encoded in the basic state. The theory is demonstrated by obtaining new results on dimension breaking of localized states for three examples: the (2+1)-Boussinesq equation, the Zakharov-Kuznetsov equation and the Kadomtsev-Petviashvili equation.
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Author(s): T.J. Bridges and G. Derks
Title: The symplectic Evans matrix and solitary wave instability
Journal: Conference proceedings Symmetry and Perturbation Theory 2001, editors: D. Bambusi, G. Gaeta and M. Cadoni, World
Scientific, pp 32-38.
YEAR: 2001
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Author(s): Grimshaw, R., Pelinovsky, D., Pelinovsky, E. and Talipova, T.
Title: Wave group dynamics in weakly nonlinear long-wave models.
Journal: Physica D, 159, 35-57.
YEAR: 2001
Author(s): Grimshaw, R.H.J., Kuznetsov, E.A., and Shapiro, E.G.
Title: The two-parameter soliton family for the interaction of a fundamental and its second harmonic.
Journal: Physica D, 152-153, 325-339.
YEAR: 2001
2000
Authors: T.J. Bridges
Title: Universal geometric condition for the transverse instability of solitary waves
Journal: Physical Review Letters 84: 2614-2617
Year: 2000
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Author(s): Clarke, S., Grimshaw, R., and Malomed, B.
Title: Soliton formation from a pulse passing the zero-dispersion point in a nonlinear Schrodinger equation.
Journal: Phys. Rev. E, 61, 5794-5801.
YEAR: 2000
1999
Author(s): G. Derks
Title: Families of relative equilibria in Hamiltonian systems with dissipation.
Journal: Conference proceedings Symmetry and Perturbation Theory 1998, Editors: A. Degasperis and G. Gaeta. World Scientific, 1999.
YEAR: 1999
Abstract: In this note the influence of dissipation on families of relative equilibria in Hamiltonian systems will be considered. Relative equilibria can be described as critical points of an appropriate functional. This characterisation can be used to give sufficient conditions such that in finite dimensional systems with dissipation the extremal families of relative equilibria are stable under dissipation. Furthermore, a full class of families of relative equilibria in the Navier-Stokes equations will be analysed. For these families it will be shown that the extremal family of relative equilibria is an attractor and the non-extremal families of relative equilibria are unstable.
Author(s): T.J. Bridges and G. Derks
Title: Unstable eigenvalues and the linearisation about solitary waves and fronts with symmetry
Journal: Proceedings of the Royal Society London, A:455, pp 2427-2469, 1999.
YEAR: 1999
Abstract: The linear stability of solitary-wave or front solutions of Hamiltonian evolutionary equations, which are equivariant with respect to a Lie group, is studied. The organizing centre for the analysis is a multi-symplectic formulation of Hamiltonian PDEs where a distinct symplectic operator is assigned for time and space. This separation of symplectic structures leads to new characterizations of the following components of the analysis. The states at infinity are characterized as manifolds of relative equilibria associated with the spatial symplectic structure. The momentum of the connecting orbit, or shape of the solitary wave, considered as a heteroclinic orbit in a phase space representation, is given a new characterization as a one-form on the tangent space to the heteroclinic manifold and this one-form is a restriction of the temporal symplectic structure. For the linear stability analysis, a new symplectic characterization of the Evan's function and its derivatives are obtained, leading to an abstract geometric proof of instability for a large class of solitary-wave states of equivariant Hamiltonian evolutionary PDEs. The theory sheds new light on several well-known models: the gKdV equation, a Boussinesq system and a nonlinear wave equation. The generalization to solitary-waves associated with multi-dimensional heteroclinic manifolds and the implications for solitary waves or fronts which are biasymptotic to invariant manifolds such as periodic states are also discussed.
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