DCDS-B

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.

DCDS

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.

DCDS

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.

DCDS-S

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.

DCDS-S

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.

DCDS-S

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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