Two-pulse solutions in the fifth-order KdV equation: Rigorous theory and numerical approximations
Marina Chugunova Dmitry Pelinovsky
Discrete & Continuous Dynamical Systems - B 2007, 8(4): 773-800 doi: 10.3934/dcdsb.2007.8.773
We revisit existence and stability of two-pulse solutions in the fifth-order Korteweg–de Vries (KdV) equation with two new results. First, we modify the Petviashvili method of successive iterations for numerical (spectral) approximations of pulses and prove convergence of iterations in a neighborhood of two-pulse solutions. Second, we prove structural stability of embedded eigenvalues of negative Krein signature in a linearized KdV equation. Combined with stability analysis in Pontryagin spaces, this result completes the proof of spectral stability of the corresponding two-pulse solutions. Eigenvalues of the linearized problem are approximated numerically in exponentially weighted spaces where embedded eigenvalues are isolated from the continuous spectrum. Approximations of eigenvalues and full numerical simulations of the fifth-order KdV equation confirm stability of two-pulse solutions associated with the minima of the effective interaction potential and instability of two-pulse solutions associated with the maxima points.
keywords: the Petviashvili method persistence and stability of nonlinear waves. the fifth-order KdV equation embedded eigenvalues two-pulse solutions
Breather continuation from infinity in nonlinear oscillator chains
Guillaume James Dmitry Pelinovsky
Discrete & Continuous Dynamical Systems - A 2012, 32(5): 1775-1799 doi: 10.3934/dcds.2012.32.1775
Existence of large-amplitude time-periodic breathers localized near a single site is proved for the discrete Klein--Gordon equation, in the case when the derivative of the on-site potential has a compact support. Breathers are obtained at small coupling between oscillators and under nonresonance conditions. Our method is different from the classical anti-continuum limit developed by MacKay and Aubry, and yields in general branches of breather solutions that cannot be captured with this approach. When the coupling constant goes to zero, the amplitude and period of oscillations at the excited site go to infinity. Our method is based on near-identity transformations, analysis of singular limits in nonlinear oscillator equations, and fixed-point arguments.
keywords: nonlocal bifurcations Discrete Klein--Gordon equation discrete breathers fixed-point arguments. asymptotic methods
Global existence of small-norm solutions in the reduced Ostrovsky equation
Roger Grimshaw Dmitry Pelinovsky
Discrete & Continuous Dynamical Systems - A 2014, 34(2): 557-566 doi: 10.3934/dcds.2014.34.557
We use a novel transformation of the reduced Ostrovsky equation to the integrable Tzitzéica equation and prove global existence of small-norm solutions in Sobolev space $H^3(\mathbb{R})$. This scenario is an alternative to finite-time wave breaking of large-norm solutions of the reduced Ostrovsky equation. We also discuss a sharp sufficient condition for the finite-time wave breaking.
keywords: Reduced Ostrovsky equation Tzitzéica equation wave breaking. conserved quantities global existence
Variational approximations of bifurcations of asymmetric solitons in cubic-quintic nonlinear Schrödinger lattices
Christopher Chong Dmitry Pelinovsky
Discrete & Continuous Dynamical Systems - S 2011, 4(5): 1019-1031 doi: 10.3934/dcdss.2011.4.1019
Using a variational approximation we study discrete solitons of a nonlinear Schrödinger lattice with a cubic-quintic nonlinearity. Using an ansatz with six parameters we are able to approximate bifurcations of asymmetric solutions connecting site-centered and bond-centered solutions and resulting in the exchange of their stability. We show that the numerical and variational approximations are quite close for solitons of small powers.
keywords: Variational approximations. Discrete nonlinear Schrödinger equations Bifurcations of discrete solitons
On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation
Tetsu Mizumachi Dmitry Pelinovsky
Discrete & Continuous Dynamical Systems - S 2012, 5(5): 971-987 doi: 10.3934/dcdss.2012.5.971
Asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation was earlier established for septic and higher-order nonlinear terms by using Strichartz estimate. We use here pointwise dispersive decay estimates to push down the lower bound for the exponent of the nonlinear terms.
keywords: Discrete nonlinear Schrödinger equation asymptotic stability. localized modes dispersive decay estimates
Dmitry Pelinovsky Milena Stanislavova Atanas Stefanov
Discrete & Continuous Dynamical Systems - S 2012, 5(5): i-iii doi: 10.3934/dcdss.2012.5.5i
Partial differential equations viewed as dynamical systems on an infinite-dimensional space describe many important physical phenomena. Lately, an unprecedented expansion of this field of mathematics has found applications in areas as diverse as fluid dynamics, nonlinear optics and network communications, combustion and flame propagation, to mention just a few. In addition, there have been many recent advances in the mathematical analysis of differential difference equations with applications to the physics of Bose-Einstein condensates, DNA modeling, and other physical contexts. Many of these models support coherent structures such as solitary waves (traveling or standing), as well as periodic wave solutions. These coherent structures are very important objects when modeling physical processes and their stability is essential in practical applications. Stable states of the system attract dynamics from all nearby configurations, while the ability to control coherent structures is of practical importance as well. This special issue of Discrete and Continuous Dynamical Systems is devoted to the analysis of nonlinear equations of mathematical physics with a particular emphasis on existence and dynamics of localized modes. The unifying idea is to predict the long time behavior of these solutions. Three of the papers deal with continuous models, while the other three describe discrete lattice equations.

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