PROC

In the present work, we introduce a new
$\mathcal{PT}$-symmetric variant of the Klein-Gordon
field theoretic problem. We identify the standing
wave solutions of the proposed class of equations
and analyze their stability. In particular, we
obtain an explicit frequency condition, somewhat
reminiscent of the classical Vakhitov-Kolokolov
criterion, which sharply separates the regimes
of spectral stability and instability. Our
numerical computations corroborate the relevant
theoretical result.

PROC

We consider the Kuramoto-Sivashinsky (KS) equation in finite domains
of the form $[-L,L]$. Our main result provides effective new estimates for higher Sobolev norms of the
solutions in terms of powers of $L$
for the one-dimentional differentiated KS. We illustrate our method on a simpler model,
namely the regularized Burger's equation.
The underlying idea in this result is that * a priori*
control of the $L^2$ norm is enough in order to conclude higher order
regularity and in fact, it allows one to get good estimates on the high-frequency tails of the solution.

DCDS

We consider the viscous Camassa-Holm equation subject to an external force,
where the viscosity term is
given by second order differential operator in divergence form. We show that under some mild assumptions on the viscosity term, one has global well-posedness both in
the periodic case and the case of the whole line. In the periodic case,
we show the existence of global attractors in the energy space $H^1$,
provided the external force is in the class $L^2(I)$. Moreover, we establish
an asymptotic smoothing effect, which states that the elements of the attractor are
in fact in the smoother Besov space B^{2} _{2, ∞}$(I)$.
Identical results (after adding an appropriate linear damping term)
are obtained in the case of the whole line.

PROC

We present
a new necessary and sufficient condition to verify
the asymptotic compactness
of an evolution equation
defined in an unbounded domain,
which involves the Littlewood-Paley projection operators.
We then use this condition to
prove the existence of an attractor
for the damped \bbme in the phase space $H^1({\bf R})$
by showing the solutions are point dissipative and asymptotically
compact. Moreover the attractor is in fact smoother and it belongs to $H^{3/2-\ve}$ for every $\ve>0$.

DCDS-B

In this paper, we study numerically the existence and stability of some special solutions of the nonlinear beam equation: $u_{tt}+u_{xxxx}+u-|u|^{p-1} u = 0$ when $p = 3$ and $p = 5$. For the standing wave solutions $u(x, t) = e^{iω t}\varphi_{ω}(x)$ we numerically illustrate their existence using variational approach. Our numerics illustrate the existence of both ground states and excited states. We also compute numerically the threshold value $ω^*$ which separates stable and unstable ground states. Next, we study the existence and linear stability of periodic traveling wave solutions $u(x, t) = φ_c(x+ct)$. We present numerical illustration of the theoretically predicted threshold value of the speed $c$ which separates the stable and unstable waves.

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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CPAA

We consider standing wave solutions of various dispersive models with non-standard form of the dispersion terms. Using index count calculations, together with the information from a variational construction, we develop sharp conditions for spectral stability of these waves.