A Renormalization Method for Modulational Stability of Quasi-Steady Patterns in Dispersive Systems

We employ global quasi-steady manifolds to rigorously reduce forced, linearly damped dispersive partial differential equations to finite dimensional flows. The manifolds we consider are not invariant, but through a renormalization group method we capture the long-time evolution of the full system as a flow on the manifold. For the parametric nonlinear Schrodinger equation we consider a manifold describing N well-separated pulses and derive an explicit system of ordinary differential equations for the flow on the manifold which captures the leading order pulse motion through the tail-tail interactions. We also outline a rigorous connection between the slow evolution in the hyperbolic PNLS and the fourth-order parabolic phase sensitive amplification equation for fiber optic systems.

[1]  J. Alexander,et al.  A topological invariant arising in the stability analysis of travelling waves. , 1990 .

[2]  E. Grenier.  Oscillatory perturbations of the Navier Stokes equations , 1997 .

[3]  E. Grenier.  Oscillatory perturbations of the Navier Stokes equations , 1997 .

[4]  J. Carr,et al.  Metastable patterns in solutions of ut = ϵ2uxx − f(u) , 1989 .

[5]  Thierry Gallay,et al.  A center-stable manifold theorem for differential equations in Banach spaces , 1993 .

[6]  Peter W. Bates,et al.  Existence and Persistence of Invariant Manifolds for Semiflows in Banach Space , 1998 .

[7]  Björn Sandstede,et al.  Existence and stability of N -pulses on optical fibers with phase-sensitive amplifiers , 1997 .

[8]  J. Bona,et al.  Higher-order asymptotics of decaying solutions of some nonlinear, dispersive, dissipative wave equations , 1995 .

[9]  Smyth,et al.  Soliton evolution and radiation loss for the nonlinear Schrödinger equation. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  C. E. Wayne,et al.  Invariant Manifolds for a Class of Dispersive, Hamiltonian, Partial Differential Equations , 1997 .

[11]  Guido Schneider,et al.  Counter-propagating Waves on Fluid Surfaces and the Continuum Limit of the Fermi-Pasta-Ulam Model , 2000 .

[12]  Peter W. Bates,et al.  Persistence of Overflowing Manifolds for Semiflow , 1999 .

[13]  Todd Kapitula,et al.  Stability of bright solitary-wave solutions to perturbed nonlinear Schro , 1998 .

[14]  Björn Sandstede,et al.  Stability of multiple-pulse solutions , 1998 .

[15]  Bifurcation and asymptotic stability in the large detuning limit of the optical parametric oscillator , 2000 .

[16]  Michael I. Weinstein,et al.  Modulational Stability of Ground States of Nonlinear Schrödinger Equations , 1985 .

[17]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.

[18]  J. M. Ball,et al.  GEOMETRIC THEORY OF SEMILINEAR PARABOLIC EQUATIONS (Lecture Notes in Mathematics, 840) , 1982 .

[19]  Tosio Kato,et al.  Quasi-linear equations of evolution, with applications to partial differential equations , 1975 .

[20]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[21]  Guido Schneider,et al.  The validity of modulation equations for extended systems with cubic nonlinearities , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[22]  C. Eugene Wayne,et al.  Invariant Manifolds for Parabolic Partial Differential Equations on Unbounded Domains , 1997 .

[23]  J. Bricmont,et al.  Renormalizing partial differential equations , 1994, chao-dyn/9411015.

[24]  R. Pego,et al.  On asymptotic stability of solitary waves , 1992 .

[25]  Peter W. Bates,et al.  Invariant Manifolds for Semilinear Partial Differential Equations , 1989 .

[26]  André Vanderbauwhede,et al.  Center Manifold Theory in Infinite Dimensions , 1992 .

[27]  J. Nathan Kutz,et al.  Stability of Pulses in Nonlinear Optical Fibers Using Phase-Sensitive Amplifiers , 1996, SIAM J. Appl. Math..

[28]  Oono,et al.  Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[29]  I. V. Barashenkov,et al.  Dynamics of the parametrically driven NLS solitons beyond the onset of the oscillatory instability , 1999 .

[30]  W. Kath,et al.  Pulse propagation in nonlinear optical fiber lines that employ phase-sensitive parametric amplifiers , 1994 .

[31]  Antti Kupiainen,et al.  Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations , 1993, chao-dyn/9306008.

[32]  Angel Sánchez,et al.  Collective Coordinates and Length-Scale Competition in Spatially Inhomogeneous Soliton-Bearing Equations , 1998, SIAM Rev..

[33]  Diana Anderson,et al.  Variational approach to nonlinear pulse propagation in optical fibers , 1983 .