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We establish local well-posedness in the Sobolev space H s with any s > 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the Camassa{Holm equation. However, unlike the more familiar Korteweg-deVries model, we demonstrate conditions on the initial data that lead to nite time blow-up of certain(More)
We investigate linear stability of solitary waves of a Hamiltonian system. Unlike weakly nonlinear water wave models, the physical system considered here is nonlinearly dispersive, and contains nonlinearity in its highest derivative term. This results in more detailed asymptotic analysis of the eigenvalue problem in presence of a large parameter. Combining(More)
We investigate how the non-analytic solitary wave solutions — peakons and compactons — of an integrable biHamiltonian system arising in fluid mechanics, can be recovered as limits of classical solitary wave solutions forming analytic homoclinic orbits for the reduced dynamical system. This phenomenon is examined to understand the important effect of linear(More)
Limitations in the current availability of bioenergy feed-stocks are a major problem in next-generation biofuels. There are global economic, political and environmental pressures to increase biofuel production and utilization, to offset gasoline and diesel fuel use, especially in the transportation sector. Many countries, such as the USA and China, have(More)
In this paper we prove the existence of self-similar solutions to the anisotropic curve shortening equation. Theorem 0.1. Given any positive C 2 function γ on S 1 there exists a solution to the equation ∂X ∂t = γ(θ)kN (0.1) which is self-similar. This means that the evolution shrinks the initial curve without changing its shape. 1 × [0, ω) → IR 2 is the(More)
We describe a pseudo-spectral numerical method to solve the systems of one-dimensional evolution equations for free surface waves in a homogeneous layer of an ideal fluid. We use the method to solve a system of one-dimensional integro-differential equations, first proposed by Ovsjannikov and later derived by Dyachenko, Zakharov, and Kuznetsov, to simulate(More)
In this part, we prove that the solitary wave solutions investigated in part I are extended as analytic functions in the complex plane, except at most countably many branch points and branch lines. We describe in detail how the limiting behavior of the complex sin-gularities allows the creation of non-analytic solutions with corners and/or compact support.(More)
We consider the linear stability and structural stability of non-ground state traveling waves of a pair of coupled nonlinear Schrr odinger equations (CNLS) which describe the evolution of co-propagating polarized pulses in the presence of birefringence. Viewing the CNLS equations as a Hamiltonian perturbation of the Manakov equations, we nd parameter(More)