A. Kairzhan, D. E. Pelinovsky, and R. H. Goodman | 2019 | Drift of spectrally stable shifted states on star graphs | arXiv.org , pp. 1902.03612 |

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Abstract:

When the coefficients of the cubic terms match the coefficients in the boundary conditions at a vertex of a star graph and satisfy a certain constraint, the nonlinear Schrödinger (NLS) equation on the star graph can be transformed to the NLS equation on a real line. Such balanced star graphs have appeared in the context of reflectionless transmission of solitary waves. Steady states on such balanced star graphs can be translated along the edges with a translational parameter and are referred to as the shifted states. When the star graph has exactly one incoming edge and several outgoing edges, the steady states are spectrally stable if their monotonic tails are located on the outgoing edges. These spectrally stable states are degenerate minimizers of the action functional with the degeneracy due to the translational symmetry. Nonlinear stability of these spectrally stable states has been an open problem up to now. In this paper, we prove that these spectrally stable states are nonlinearly unstable because of the irreversible drift along the incoming edge towards the vertex of the star graph. When the shifted states reach the vertex as a result of the drift, they become saddle points of the action functional, in which case the nonlinear instability leads to their destruction. In addition to rigorous mathematical results, we use numerical simulations to illustrate the drift instability and destruction of the shifted states on the balanced star graph.

BibTeX:

@article{driftStates,

Author = {A. Kairzhan, D. E. Pelinovsky, R. H. Goodman},

Title = {Drift of spectrally stable shifted states on star graphs},

Journal = {arXiv.org},

Volume = {},

Pages = {1902.03612},

Year = {2019}

}

Notes:

R. H. Goodman | 2019 | NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph | Disc. Cont. Dyn. Sys. A 39, pp. 2203--2232 |

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We consider the bifurcations of standing wave solutions to the nonlinear Schrödinger equation (NLS) posed on a quantum graph consisting of two loops connected by a single edge, the so-called dumbbell, recently studied by Marzuola and Pelinovsky. The authors of that study found the ground state undergoes two bifurcations, first a symmetry-breaking, and the second which they call a symmetry-preserving bifurcation. We clarify the type of the symmetry-preserving bifurcation, showing it to be transcritical. We then reduce the question, and show that the phenomena described in that paper can be reproduced in a simple discrete self-trapping equation on a combinatorial graph of bowtie shape. This allows for complete analysis by parameterizing the full solution space. We then expand the question, and describe the bifurcations of all the standing waves of this system, which can be classified into three families, and of which there exists a countably infinite set.

BibTeX:

@article{bowtie,

Author = {R. H. Goodman},

Title = {NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph},

Journal = {Disc. Cont. Dyn. Sys. A},

Volume = {39},

Pages = {2203--2232},

Year = {2019}

}

Notes:

R. H. Goodman | 2017 | Bifurcations of relative periodic orbits in NLS/GP with a triple-well potential | Phys. D 359, pp. 39--59 |

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The nonlinear Schrödinger/Gross--Pitaevskii (NLS/GP) equation is considered in the presence of three equally-spaced potentials. The problem is reduced to a finite-dimensional Hamiltonian system by a Galerkin truncation. Families of oscillatory orbits are sought in the neighborhoods of the system's nine branches of standing wave solutions. Normal forms are computed in the neighborhood of these branches' various Hamiltonian Hopf and saddle--node bifurcations, showing how the oscillatory orbits change as a parameter is increased. Numerical experiments show agreement between normal form theory and numerical solutions to the reduced system and NLS/GP near the Hamiltonian Hopf bifurcations and some subtle disagreements near the saddle--node bifurcations due to exponentially small terms in the asymptotics.

BibTeX:

@article{GoodmanPhysD2017,

Author = {R. H. Goodman},

Title = {Bifurcations of relative periodic orbits in NLS/GP with a triple-well potential},

Journal = {Phys. D},

Volume = {359},

Pages = {39--59},

Year = {2017}

}

Notes:

R. H. Goodman, A. Rahman, M. Bellanich, and C. N. Morrison | 2015 | A Mechanical Analog of the Two-Bounce Resonance of Solitary Waves: Modeling and Experiment | Chaos 25, pp. 043109 |

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We describe a simple mechanical system, a ball rolling along a specially-designed landscape, that mimics the dynamics of a well known phenomenon, the two-bounce resonance of solitary wave collisions, that has been seen in countless numerical simulations but never in the laboratory. We provide a brief history of the solitary wave problem, stressing the fundamental role collective-coordinate models played in understanding this phenomenon. We derive the equations governing the motion of a point particle confined to such a surface and then design a surface on which to roll the ball, such that its motion will evolve under the same equations that approximately govern solitary wave collisions. We report on physical experiments, carried out in an undergraduate applied mathematics course, that seem to verify one aspect of chaotic scattering, the so-called two-bounce resonance.

BibTeX:

@article{skewball,

Author = {R. H. Goodman, A. Rahman, M. Bellanich, C. N. Morrison},

Title = {A Mechanical Analog of the Two-Bounce Resonance of Solitary Waves: Modeling and Experiment},

Journal = {Chaos},

Volume = {25},

Pages = {043109},

Year = {2015}

}

Notes:

This was the principal experiment in our 2010 capstone class.
For more about NJIT's applied math capstone class, see here.
Animated version of Figure 7.

R. H. Goodman, P. G. Kevrekidis, and R. Carretero-González | 2015 | Dynamics of vortex dipoles in anisotropic Bose-Einstein condensates | SIAM J. Appl. Dyn. Sys. , pp. 699-729 |

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Abstract:

We study the motion of a vortex dipole in a Bose-Einstein condensate confined
to an anisotropic trap. We focus on a system of ordinary
differential equations describing the vortices' motion, which is in turn a reduced model
of the Gross-Pitaevskii equation describing the condensate's motion.
Using a sequence of canonical changes of variables, we reduce the
dimension and simplify the equations of motion. We uncover two interesting
regimes. Near a family of periodic orbits known as guiding centers, we find
that the dynamics is essentially that of a pendulum coupled to
a linear oscillator, leading to stochastic reversals in the overall
direction of rotation of the dipole. Near the separatrix orbit in the
isotropic system, we find other families of periodic, quasi-periodic,
and chaotic trajectories. In a neighborhood of the guiding
center orbits, we derive an explicit iterated map that simplifies the
problem further. Numerical calculations are used to illustrate the
phenomena discovered through the analysis. Using the results from the
reduced system we are able to construct complex periodic orbits in
the original, partial differential equation, mean-field model for
Bose-Einstein condensates, which corroborates the phenomenology
observed in the reduced dynamical equations.

BibTeX:

@article{AnisotropicBEC2014,

Author = {R. H. Goodman, P. G. Kevrekidis, R. Carretero-González},

Title = {Dynamics of vortex dipoles in anisotropic Bose-Einstein condensates},

Journal = {SIAM J. Appl. Dyn. Sys.},

Volume = {},

Pages = {699-729},

Year = {2015}

}

Notes:

R. H. Goodman, J. L. Marzuola, and M. I. Weinstein | 2015 | Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii Equation | Disc. Cont. Dyn. Sys. A 35, pp. 225--246 |

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We study the long-time behavior of solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation (NLS/GP) with a symmetric double-well potential. NLS/GP governs nearly-monochromatic guided optical beams in weakly coupled waveguides with both linear and nonlinear (Kerr) refractive indices and zero absorption, as well as the behavior of Bose-Einstein condensates. For small L2 norm (low power), the solution executes beating oscillations between the two wells. There is a power threshold at which a symmetry breaking bifurcation occurs. The set of guided mode solutions splits into two families of solutions. One type of solution is concentrated in either well of the potential, but not both. Solutions in the second family undergo tunneling oscillations between the two wells. A finite dimensional reduction (system of ODEs) derived in Marzuola and Weinstein 2010 and is expected to well-approximate the PDE dynamics on long time scales. In particular, we revisit this reduction, find a class of exact solutions and shadow them in the (NLS/GP) system by applying the above approach.

BibTeX:

@article{gmw_2015,

Author = {R. H. Goodman, J. L. Marzuola, M. I. Weinstein},

Title = {Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii Equation},

Journal = {Disc. Cont. Dyn. Sys. A},

Volume = {35},

Pages = {225--246},

Year = {2015}

}

Notes:

This extends the result of Marzuola and M. I. Weinstein. **Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations**. DCDS-A, 28:1505--1554, 2010.
There, they proved that small amplitude solutions to a reduced model equation, essentially the double-well Duffing oscillator are shadowed by nearby solutions to NLS/GP. Here, we were able to write down the general solution to the ODE system and thus prove a similar theorem about the shadowing of more general ODE solutions by PDE solutions.

J. K. Wróbel and R. H. Goodman | 2013 | High-order Adaptive Method for Computing Two-dimensional Invariant Manifolds of Maps | Commun. Nonlinear Sci. 18, pp. 1734-1745 |

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An efficient and accurate numerical method is presented for computing invariant manifolds of maps which arise in the study of dynamical systems. A quasi-interpolation method due to Hering-Bertram et al. is used to decrease the number of points needed to compute a portion of the manifold. Bézier triangular patches are used in this construction, together with adaptivity conditions based on properties of these patches. Several numerical tests are performed, which show the method to compare favorably with previous approaches.

BibTeX:

@article{CNSNS_2013,

Author = {J. K. Wróbel, R. H. Goodman},

Title = {High-order Adaptive Method for Computing Two-dimensional Invariant Manifolds of Maps},

Journal = {Commun. Nonlinear Sci.},

Volume = {18},

Pages = {1734-1745},

Year = {2013}

}

Notes:

The second half of Jacek's thesis.

R. H. Goodman | 2011 | Hamiltonian Hopf bifurcations and dynamics of NLS/GP standing-wave modes | J. Phys. A: Math. Theor. 44, pp. 425101 |

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We examine the dynamics of solutions to nonlinear Schrödinger/Gross-- Pitaevskii equations that arise due to semisimple indefinite Hamiltonian Hopf bifurcations---the collision of pairs of eigenvalues on the imaginary axis. We construct localized potentials for this model which lead to such bifurcations in a predictable manner. We perform a formal reduction from the partial differential equations to a small system of ordinary differential equations. We analyze the equations to derive conditions for this bifurcation and use averaging to explain certain features of the dynamics that we observe numerically. A series of careful numerical experiments are used to demonstrate the phenomenon and the relations between the full system and the derived approximations.

BibTeX:

@article{JPhysA2011,

Author = {R. H. Goodman},

Title = {Hamiltonian Hopf bifurcations and dynamics of NLS/GP standing-wave modes},

Journal = {J. Phys. A: Math. Theor.},

Volume = {44},

Pages = {425101},

Year = {2011}

}

Notes:

I think there is a lot to be learned about optical systems by computing normal forms in this sort of bifurcation problem.

I was asked to write a short summary for their Insights page.

**Erratum:** At the end of section 2.4.3, top of page 10, I state that "a straightforward calculation" shows that the HH bifurcation studied by Johansson for the periodic NLS trimer to come from a semisimple -1:1 resonance. In fact, it is the generic non-semisimple bifurcation.

R. H. Goodman and J. K. Wróbel | 2011 | High-Order Bisection Method for Computing Invariant Manifolds of Two-Dimensional Maps | Int. J. Bifurcat. Chaos 21, pp. 2017-2042 |

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BibTeX:

@article{IJBC2011,

Author = {R. H. Goodman, J. K. Wróbel},

Title = {High-Order Bisection Method for Computing Invariant Manifolds of Two-Dimensional Maps},

Journal = {Int. J. Bifurcat. Chaos},

Volume = {21},

Pages = {2017-2042},

Year = {2011}

}

Notes:

The first paper to come out of work with Jacek.

J. Bławzdziewicz, R. H. Goodman, N. Khurana, E. Wajnryb, and Y.-N. Young | 2010 | Nonlinear hydrodynamic phenomena in Stokes flow regime | Phys. D 239, pp. 1214 - 1224 |

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We investigate nonlinear phenomena in dispersed two-phase systems under creeping-flow conditions. We consider nonlinear evolution of a single deformed drop and collective dynamics of arrays of hydrodynamically coupled particles. To explore physical mechanisms of system instabilities, chaotic drop evolution, and structural transitions in particle arrays we use simple models, such as small-deformation equations and effective-medium theory. We find numerical and analytical solutions of the simplified governing equations. The small-deformation equations for drop dynamics are analyzed using results of dynamical systems theory. Our investigations shed new light on the dynamics of complex fluids, where the nonlinearity often stems from the evolving boundary conditions in Stokes flow.

BibTeX:

@article{Blawzdziewicz20101214,

Author = {J. Bławzdziewicz, R. H. Goodman, N. Khurana, E. Wajnryb, Y.-N. Young},

Title = {Nonlinear hydrodynamic phenomena in Stokes flow regime},

Journal = {Phys. D},

Volume = {239},

Pages = {1214 - 1224},

Year = {2010}

}

Notes:

Y.-N. Young, J. Bławzdziewicz, V. Cristini, and R. H. Goodman | 2008 | Hysteretic and chaotic dynamics of viscous drops in creeping flows with rotation | J. Fluid. Mech 607, pp. 209--234 |

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BibTeX:

@article{JFM08,

Author = {Y.-N. Young, J. Bławzdziewicz, V. Cristini, R. H. Goodman},

Title = {Hysteretic and chaotic dynamics of viscous drops in creeping flows with rotation},

Journal = {J. Fluid. Mech},

Volume = {607},

Pages = {209--234},

Year = {2008}

}

Notes:

R. H. Goodman | 2008 | Chaotic scattering in solitary wave interactions: A singular iterated-map description | Chaos 18, pp. 023113 |

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BibTeX:

@article{Chaos2008,

Author = {R. H. Goodman},

Title = {Chaotic scattering in solitary wave interactions: A singular iterated-map description},

Journal = {Chaos},

Volume = {18},

Pages = {023113},

Year = {2008}

}

Notes:

This is the paper that really, I think, cracks open the n-bounce resonance/chaotic scattering of solitons problem. The iterated map description whittles away the extraneous details and lays bare the mathematical structure in a way that our previous papers have not.
Copyright 2008 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.

R. H. Goodman and M. I. Weinstein | 2008 | Stability and instability of nonlinear defect states in the coupled mode equations---analytical and numerical study | Phys. D. 237, pp. 2731-2760 |

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BibTeX:

@article{GW_PHYSD_2008,

Author = {R. H. Goodman, M. I. Weinstein},

Title = {Stability and instability of nonlinear defect states in the coupled mode equations---analytical and numerical study},

Journal = {Phys. D.},

Volume = {237},

Pages = {2731-2760},

Year = {2008}

}

Notes:

R. H. Goodman and R. Haberman | 2007 | Chaotic Scattering and the n-bounce Resonance in Solitary Wave Interactions | Phys. Rev. Lett. 98, pp. 104103 |

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We present a new and complete analysis of the n-bounce resonance and chaotic scattering in solitary wave collisions. In these phenomena, the speed at which a wave exits a collision depends in a complicated fractal way on its input speed. We present a new asymptotic analysis of collective-coordinate ODEs, reduced models that reproduce the dynamics of these systems. We reduce the ODEs to discrete-time iterated separatrix maps and obtain new quantitative results unraveling the fractal structure of the scattering behavior. These phenomena have been observed repeatedly in many solitary-wave systems over 25 years.

BibTeX:

@article{GH_PRL_07,

Author = {R. H. Goodman, R. Haberman},

Title = {Chaotic Scattering and the n-bounce Resonance in Solitary Wave Interactions},

Journal = {Phys. Rev. Lett.},

Volume = {98},

Pages = {104103},

Year = {2007}

}

Notes:

This is a summary and a tying-together of the methods used in the three earlier papers with Haberman. As most papers in PRL, I imagine this paper is hard to read unless you're willing to do all the math again yourself. Most of these ideas are presented in more detail in the paper in SIAM Journal of Applied Dynamical Systems. The unnumbered equation before (19) is based on a recent observation by Rich Haberman. We can generalize equation (19) to four or five-bounce resonant solutions, but the space limitations for PRL precluded showing this.

Writing a PRL summarizing these ideas was encouraged by Alwyn Scott, who also gave many suggestions for improvements when the paper was initially rejected, and who passed away before it was published. We asked and, unfortunately, the PRL style manual does not allow for dedications. Nonetheless, this paper is dedicated to Al Scott.

This paper was written up by David Campbell in a "Journal Club" column at the front of Nature, see here.

R. H. Goodman and R. Haberman | 2005 | Vector soliton interactions in birefringent optical fibers | Phys. Rev. E 71, pp. 056605 |

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We consider the interactions of two identical, orthogonally polarized vector solitons in a nonlinear optical fiber with two polarization directions, described by a coupled pair of nonlinear Schroedinger equations. We study a low-dimensional model system of Hamiltonian ordinary differential equations (ODEs) derived by Ueda and Kath and also studied by Tan and Yang. We derive a further simplified model which has similar dynamics but is more amenable to analysis. Sufficiently fast solitons move by each other without much interaction, but below a critical velocity the solitons may be captured. In certain bands of initial velocities the solitons are initially captured, but separate after passing each other twice, a phenomenon known as the two-bounce or two-pass resonance. We derive an analytic formula for the critical velocity. Using matched asymptotic expansions for separatrix crossing, we determine the location of these ``resonance windows.'' Numerical simulations of the ODE models show they compare quite well with the asymptotic theory.

BibTeX:

@article{GH_PRE_05,

Author = {R. H. Goodman, R. Haberman},

Title = {Vector soliton interactions in birefringent optical fibers},

Journal = {Phys. Rev. E},

Volume = {71},

Pages = {056605},

Year = {2005}

}

Notes:

This problem makes use of a curious fact that lots of people use but nobody, as far as I know, can explain. Namely, in many situations where the averaged Lagrangian is used to derive ODE's for some perturbation of cubic NLS, it is necessary to include a degree of freedom which represents an oscillation between the solitons' widths and their chirps. If you don't include these terms, it doesn't work. Other people have seen this in related problems. This is strange. The other time-dependent quantities all arise from the symmetries of the underlying PDE (position, phase, velocity, total intensity). What is the mathematical importance of the chirp/width degree of freedom and where does it come from?

In this paper we made the approximation that the secondary oscillator is linear. Using ideas from Camassa, Kovacic, and Tin, it may be possible to remove this approximation by putting the secondary oscillator in action-angle coordinates. Note this approximation does not seem to effect the qualitative structure of the resonace windows, but it has a big effect on the fine details, compare figures 3 and 6.

R. H. Goodman and R. Haberman | 2005 | Kink-antikink collisions in the phi-four equation: The n-bounce resonance and the separatrix map | SIAM J. Appl. Dyn. Sys. 4, pp. 1195-1228 |

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We provide a detailed mathematical explanation of a phenomenon known as the two-bounce resonance observed in collisions between kink and anti-kink traveling waves of the phi-four equations of mathematical physics. This behavior was discovered numerically in the 1980's by Campbell and his collaborators and subsequently discovered in several other equations supporting traveling waves. We first demonstrate the effect with new high-resolution numerical simulations. A pair of kink-like traveling waves may coalesce into a localized bound state or may reflect off each other. In the two bounce-resonance, they first coalesce, but later escape each others' embrace, with a very regular pattern governing the behaviors. Studying a finite-dimensional "collective coordinates" model, we use geometric phase-plane based reasoning and matched asymptotics to explain the mechanism underlying the phenomenon, including the origin of several mathematical assumptions needed by previous researchers. We derive a separatrix map for this problem---a simple algebraic recursion formula that explains the complex fractal-like dependence on initial velocity for kink-antikink interactions.

BibTeX:

@article{GH_SIADS_05,

Author = {R. H. Goodman, R. Haberman},

Title = {Kink-antikink collisions in the phi-four equation: The n-bounce resonance and the separatrix map},

Journal = {SIAM J. Appl. Dyn. Sys.},

Volume = {4},

Pages = {1195-1228},

Year = {2005}

}

Notes:

I had wanted to send the Physica D paper that Haberman and I wrote to SIADS, but Phil Holmes, who is on the editorial board, told me he thought papers without impressive graphics or animations would have a hard time getting published there. When I was working on this paper, I wanted to have nice color graphs for SIADS, and came up with figure 13, which led me to figures 14 and 15. Seeing these figures drastically shifted the way I think about these problems and to what I think is the first real understanding of the mechanisms behind chaotic scattering in all the n-bounce type problems.

R. H. Goodman, R. E. Slusher, M. I. Weinstein, and M. Klaus | 2005 | Trapping light with grating defects | Contemporary Mathematics 379: Mathematical studies in nonlinear wave propagation , pp. 83-92 |

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Gap solitons are localized traveling waves that exist in Bragg grating optical fibers. We demonstrate a family of grating defects that support linear and nonlinear standing wave modes, and show numerically that these defect modes may be used to trap the energy from a gap soliton. A mechanism involving a nonlinear resonance is proposed to explain why trapping occurs in some situations and not in others.

BibTeX:

@article{GSWK_05,

Author = {R. H. Goodman, R. E. Slusher, M. I. Weinstein, M. Klaus},

Title = {Trapping light with grating defects},

Journal = {Contemporary Mathematics 379: Mathematical studies in nonlinear wave propagation},

Volume = {},

Pages = {83-92},

Year = {2005}

}

Notes:

Although this proceedings article is mainly a short summary of the JOSA-B paper with Slusher and Weinstein, it contains one significant new result. We derive the eigenfunctions and eigenvalues for the defects in the linear limit, confirming formulas we had determined via very accurate numerics in the JOSA-B paper. Martin Klaus showed us how to use a supersymmetry argument to calculate these.

R. H. Goodman and R. Haberman | 2004 | Interaction of sine-Gordon kinks with defects: The two-bounce resonance | Phys. D. 195, pp. 303--323 |

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A model of soliton--defect interactions in the sine-Gordon equations is studied using singular perturbation theory. Melnikov theory is used to derive a critical velocity for strong interactions, which is shown to be exponentially small for weak defects. Matched asymptotic expansions for nearly heteroclinic orbits are constructed for the initial value problem, which are then used to derive analytical formulas for the locations of the well known two- and three-bounce resonance windows, as well as several other phenomena seen in numerical simulations.

BibTeX:

@article{GH_PhysD_04,

Author = {R. H. Goodman, R. Haberman},

Title = {Interaction of sine-Gordon kinks with defects: The two-bounce resonance},

Journal = {Phys. D.},

Volume = {195},

Pages = {303--323},

Year = {2004}

}

Notes:

The first paper Haberman and I wrote about this subject. In later papers we were able to simplify and clarify the main calculation by integrating the last equation on page 310 by parts. Compare with equation (22) of the SIADS paper "Kink-antikink collision..." The presentation is somewhat clearer and better organized in that paper, although the equations in this paper are much simpler.

R. H. Goodman, P. J. Holmes, and M. I. Weinstein | 2004 | Strong NLS soliton--defect interactions | Phys. D 192, pp. 215--248 |

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We consider the interaction of a nonlinear Schro?dinger soliton with a spatially localized (point) defect in the medium through which it travels. Using numerical simulations, we find parameter regimes under which the soliton may be reflected, transmitted, or captured by the defect. We propose a mechanism of resonant energy transfer to a nonlinear standing wave mode supported by the defect. Extending Forinash et al. [Phys. Rev. E 49 (1994) 3400], we then derive a finite-dimensional model for the interaction of the soliton with the defect via a collective coordinates method. The resulting system is a three degree-of-freedom Hamiltonian with an additional conserved quantity. We study this system both numerically and using the tools of dynamical systems theory, and find that it exhibits a variety of interesting behaviors, largely determined by the structures of stable and unstable manifolds of special classes of periodic orbits. We use this geometrical understanding to interpret the simulations of the finite-dimensional model, compare them with the nonlinear Schr\"odinger simulations, and comment on differences due to the finite-dimensional ansatz.

BibTeX:

@article{GHW_PhysD_04,

Author = {R. H. Goodman, P. J. Holmes, M. I. Weinstein},

Title = {Strong NLS soliton--defect interactions},

Journal = {Phys. D},

Volume = {192},

Pages = {215--248},

Year = {2004}

}

Notes:

Related to this paper are some very beautiful results, both rigorous and numeric, by Holmer, Marzuola, and Zworski at Berkeley.

R. H. Goodman, P. J. Holmes, and M. I. Weinstein | 2002 | Interaction of sine-Gordon kinks with defects: phase space transport in a two-mode model | Phys. D 161, pp. 21--44 |

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We study a model derived by Fei et al. [Phys. Rev. A 45 (1992) 6019] of a kink solution to the sine-Gordon equation interacting with an impurity mode. The model is a two degree of freedomHamiltonian system. We investigate this model using the tools of dynamical systems, and show that it exhibits a variety of interesting behaviors including transverse heteroclinic orbits to degenerate equilibria at infinity, chaotic dynamics, and an extremely complex and delicate structure describing the interaction of the kink with the defect. We interpret this in terms of phase space transporttheory.

BibTeX:

@article{GHW_PhysD_02,

Author = {R. H. Goodman, P. J. Holmes, M. I. Weinstein},

Title = {Interaction of sine-Gordon kinks with defects: phase space transport in a two-mode model},

Journal = {Phys. D},

Volume = {161},

Pages = {21--44},

Year = {2002}

}

Notes:

I returned to the subject of this paper again in the Physica D paper with Haberman, where we significantly strengthened the results.

R. H. Goodman, R. E. Slusher, and M. I. Weinstein | 2002 | Stopping Light on a Defect | J. Opt. Soc. Am. B 19, pp. 1635--1632 |

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Gap solitons are localized nonlinear coherent states that have been shown both theoretically and experimentally to propagate in periodic structures. Although theory allows for their propagation at any speed v, between 0 and c, they have been observed in experiments at speeds of approximately 50% of c. It is of scientific and technological interest to trap gap solitons. We first introduce an explicit multiparameter family of periodic structures with localized defects, which support linear defect modes. These linear defect modes are shown to persist into the nonlinear regime, as nonlinear defect modes. Using mathematical analysis and numerical simulations, we then investigate the capture of an incident gap soliton by these defects. The mechanism of capture of a gap soliton is resonant transfer of its energy to nonlinear defect modes. We introduce a useful bifurcation diagram from which information on the parameter regimes of gap-soliton capture, reflection, and transmission can be obtained by simple conservation of energy and resonant energy transfer principles.

BibTeX:

@article{GSW_JOSAB_02,

Author = {R. H. Goodman, R. E. Slusher, M. I. Weinstein},

Title = {Stopping Light on a Defect},

Journal = {J. Opt. Soc. Am. B},

Volume = {19},

Pages = {1635--1632},

Year = {2002}

}

Notes:

The publisher, despite numerous complaints and attempts on our part to correct them, could not get the citation numbering to line up properly. We managed to get this corrected in this errata.

P. J. Holmes, R. H. Goodman, and M. I. Weinstein | 2002 | Trapping of kinks and solitons by defects: phase space transport in finite-dimensional models | Progress in nonlinear science, Vol. 1 (Nizhny Novgorod, 2001) RAS, Inst. Appl. Phys., Nizhni Novgorod, pp. 103--115 |

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We study models of Fei et al. and of Forinash et al of kinks in the sine-Gordon equation, and solitons in the nonlinear Schrodinger equation interacting with point defects. The models are two and three degree-of-freedom Hamiltonian systems. Using dynamical systems methods, we show that they exhibit interesting behaviors including transverse heteroclinic orbits to degenerate equilibria at infinity, chaotic dynamics and complex and delicate structures describing the interaction of traveling waves with the defect. We interpret the behavior in terms of invariant manifolds and phase space transport theory.

BibTeX:

@article{HGW,

Author = {P. J. Holmes, R. H. Goodman, M. I. Weinstein},

Title = {Trapping of kinks and solitons by defects: phase space transport in finite-dimensional models},

Journal = {Progress in nonlinear science, Vol. 1 (Nizhny Novgorod, 2001)},

Volume = {RAS, Inst. Appl. Phys., Nizhni Novgorod},

Pages = {103--115},

Year = {2002}

}

Notes:

This is a proceedings article, which is superceded by later publications with the same collaborators and with Haberman.

R. H. Goodman, M. I. Weinstein, and P. J. Holmes | 2001 | Nonlinear propagation of light in one-dimensional periodic structures | J. Nonlinear Science 11, pp. 123--168 |

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Abstract:

We consider the nonlinear propagation of light in an optical fiber waveguide as modeled by the anharmonic Maxwell-Lorentz equations (AMLE). The waveguide is assumed to have an index of refraction that varies periodically along its length. The wavelength of light is selected to be in resonance with the periodic structure (Bragg resonance). The AMLE system considered incorporates the effects of noninstantaneous response of the medium to the electromagnetic field (chromatic or material dispersion), the periodic structure (photonic band dispersion), and nonlinearity. We present a detailed discussion of the role of these effects individually and in concert. We derive the nonlinear coupled mode equations (NLCME) that govern the envelope of the coupled backward and forward components of the electromagnetic field. We prove the validity of the NLCME description and give explicit estimates for the deviation of the approximation given by NLCME from the exact dynamics, governed by AMLE. NLCME is known to have gap soliton states. A consequence of our results is the existence of very long-lived gap soliton states of AMLE. We present numerical simulations that validate as well as illustrate the limits of the theory. Finally, we verify that the assumptions of our model apply to the parameter regimes explored in recent physical experiments in which gap solitons were observed.

BibTeX:

@article{GWH_JNLS_01,

Author = {R. H. Goodman, M. I. Weinstein, P. J. Holmes},

Title = {Nonlinear propagation of light in one-dimensional periodic structures},

Journal = {J. Nonlinear Science},

Volume = {11},

Pages = {123--168},

Year = {2001}

}

Notes:

R. H. Goodman, A. J. Majda, and D. W. McLaughlin | 2001 | Modulations in the leading edges of midlatitude storm tracks | SIAM J. Appl. Math. 62, pp. 746--776 |

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Abstract:

Downstream development is a term encompassinga variety of effects relatingto the propagation of storm systems at midlatitude. We investigate a mechanism behind downstream development and study how wave propagation is affected by varying several physical parameters. We then develop a multiple scales modulation theory based on processes in the leadingedge of propagating fronts to examine the effect of nonlinearity and weak variation in the background flow. Detailed comparisons are made with numerical experiments for a simple model system.

BibTeX:

@article{GMM_SIAPL_01,

Author = {R. H. Goodman, A. J. Majda, D. W. McLaughlin},

Title = {Modulations in the leading edges of midlatitude storm tracks},

Journal = {SIAM J. Appl. Math.},

Volume = {62},

Pages = {746--776},

Year = {2001}

}

Notes:

R. H. Goodman, D. S. Graff, L. M. Sander, P. Leroux-Hugon, and E. Clement | 1995 | Trigger Waves in a Model for Catalysis | Phys. Rev. E 52, pp. 5904--5909 |

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Abstract:

We consider the model of catalysis due to Ziff, Gulari, and Barshad [Phys Rev. Lett. 56, 2553 (1986)] as a pattern formation problem. We find that the model supports rigger waves and we examine the dependence of the wave velocity on diffusion. In addition to the usual interface width there is a statistical broadening of the wave front that increases in time at t^{1/3}.

BibTeX:

@article{triggerwaves,

Author = {R. H. Goodman, D. S. Graff, L. M. Sander, P. Leroux-Hugon, E. Clement},

Title = {Trigger Waves in a Model for Catalysis},

Journal = {Phys. Rev. E},

Volume = {52},

Pages = {5904--5909},

Year = {1995}

}

Notes:

Created on 2019-02-19 09:41:54 +0000 using BibDesk for the Mac, based on an export template by Mauro Cherubini

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