Computing the Maslov index of solitary waves. I: Hamiltonian systems on a four-dimensional phase space

DOI10.1016/j.physd.2009.05.008zbMath1179.37097OpenAlexW2116735602MaRDI QIDQ732347

Thomas J. Bridges, Frédéric Dias, Frédéric Chardard

Publication date: 9 October 2009

Published in: Physica D (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.physd.2009.05.008

zbMATH Keywords

stabilitysolitary wavesEvans functionHamiltonian systemsMaslov indexlong-wave-short-wave resonance

Mathematics Subject Classification ID

KdV equations (Korteweg-de Vries equations) (35Q53) Reaction-diffusion equations (35K57) NLS equations (nonlinear Schrödinger equations) (35Q55) Soliton equations (35Q51) Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems (37K40)

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Cites Work

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• Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework
• Computing the Maslov index of solitary waves. I: Hamiltonian systems on a four-dimensional phase space
• Stability of solitary waves of a fifth-order water wave model
• Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics
• The Sturm theorems and symplectic geometry
• Chaos in classical and quantum mechanics
• Homoclinic period blow-up in reversible and conservative systems
• On the Morse index in variational calculus
• The Maslov index for paths
• Path integration over closed loops and Gutzwiller's trace formula
• Numerical exterior algebra and the compound matrix method
• Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system
• On the nonlinear stability of solitary wave solutions of the fifth-order Korteweg-de Vries equation
• On the stability of solitary wave solutions of the fifth-order KdV equation.
• Maslov index for solitary waves obtained as a limit of the Maslov index for periodic waves
• Two-pulse solutions in the fifth-order KdV equation: rigorous theory and numerical approximations
• Maslov index for homoclinic orbits of Hamiltonian systems
• A note on the geometry of linear Hamiltonian systems of signature 0 in \(\mathbb R^{4}\)
• Maslov indices in the Gutzwiller trace formula
• Fast computation of the Maslov index for hyperbolic linear systems with periodic coefficients
• On the Maslov index of multi-pulse homoclinic orbits
• Instability of standing waves for non-linear Schrödinger-type equations
• On the Multi-soliton Solutions of Some Nonlinear Evolution Equations
• On the canonically invariant calculation of Maslov indices
• Constructing the symplectic Evans matrix using maximally analytic individual vectors
• Winding number formula for Maslov indices
• Eigenvalues, and instabilities of solitary waves
• Linear Instability of Solitary Wave Solutions of the Kawahara Equation and Its Generalizations
• The Spectral Flow and the Maslov Index
• Stability of the in-phase travelling wave solutions in a pair of coupled nerve fibers
• Spectral stability and time evolution ofN-solitons in the KdV hierarchy
• DIFFERENTIAL GEOMETRY OF GRASSMANN MANIFOLDS

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