Normal form for travelling kinks in discrete Klein-Gordon lattices

We study travelling kinks in the spatial discretizations of the nonlinear Klein--Gordon equation, which include the discrete $\phi^4$ lattice and the discrete sine--Gordon lattice. The differential advance-delay equation for travelling kinks is reduced to the normal form, a scalar fourth-order differential equation, near the quadruple zero eigenvalue. We show numerically non-existence of monotonic kinks (heteroclinic orbits between adjacent equilibrium points) in the fourth-order equation. Making generic assumptions on the reduced fourth-order equation, we prove the persistence of bounded solutions (heteroclinic connections between periodic solutions near adjacent equilibrium points) in the full differential advanced-delay equation with the technique of center manifold reduction. Existence and persistence of multiple kinks in the discrete sine--Gordon equation are discussed in connection to recent numerical results of \cite{ACR03} and results of our normal form analysis

Existence of Bifurcating Quasipatterns in Steady Bénard–Rayleigh Convection

Extending the results obtained in the sixties for bifurcating periodic patterns, the existence of bifurcating quasipatterns in the steady Benard-Rayleigh convection problem is proved. These are two-dimensional patterns, quasiperiodic in any horizontal direction, invariant under horizontal rotations of angle /q. There is a small divisor problem for q4.Using the results of Berti-Bolle-Procesi in 2010, we adapt it to a Navier-Stokes system ruling the Benard-Rayleigh convection problem. Our solution is approximated by the truncated power series which was formally obtained by Iooss in 2009, but which is divergent in general (Gevrey series). First, we formulate the problem in introducing a suitable parameter, able to move the spectrum of the linearized operator, as a whole, as for the Swift-Hohenberg PDE model. For using the Nash-Moser process, we are faced with the problem of inverting a linear operator which is the differential at a non zero point.There are two new difficulties: (i) First, the extra dimension leading to a more complicated spectrum of the linear operator. This first difficulty leads to use specific projections for reducing the spectrum of the studied operator, which we want to invert, to a finite set very close to 0. (ii) The second difficulty is the fact that the linearization L-(N) at a non-zero point leads to a non-selfadjoint operator, contrary to what occurs in previous works. This is more serious, and leads to use the spectrum of (LL(N)*)-L-(N) which depends mainly quadratically on the main parameter. A careful study of the bad setof parameters, with an assumption on the convexity of the eigenvalues of this operator, allows us to obtain a good estimate, as it is necessary for using the results of Berti etal. for solving the range equation. We again use separation properties of the Fourier spectrum (see the Bourgain and Craig results) for obtaining an estimate in high Sobolev norms. It then remains to solve the one-dimensional bifurcation equation.For any q4 , and provided that a weak transversality conjecture is realized, we prove the existence of a bifurcating convective quasipattern of order 2q, above the critical Rayleigh number

Proof of Quasipatterns for the Swift-Hohenberg Equation

This paper establishes the existence of quasipatterns solutions of the Swift-Hohenberg PDE. In a former approach we avoided the use of Nash-Moser scheme, but our proof contains a gap. The present proof of existence is based on the works by Berti et al related to the Nash-Moser scheme. For solving the small divisor problem, we need to introduce a new free parameter related to the freedom in the choice of parameterization of the bifurcating solution. Thanks to a transversality condition, the result gives only a bifurcating set, located in a small hornlike region centered on a curve, with the origin at the bifurcation point

Solitary waves of a coupled Korteweg-de Vries system

In the long-wave, weakly nonlinear limit a generic model for the interaction of two waves with nearly coincident linear phase speeds is a pair of coupled Korteweg-de Vries equations. Here we consider the simplest case when the coupling occurs only through linear non-dispersive terms, and for this case delineate the various families of solitary waves that can be expected. Generically, we demonstrate that there will be three families, (a) pure solitary waves which decay to zero at in nity exponentially fast, (b) generalized solitary waves which may tend to small-amplitude oscillations at in nity, and (c) envelope solitary waves which at in nity consist of decaying oscillations. We use a combination of asymptotic methods and the rigorous results obtained from a normal form approach to determine these solitary wave families

Existence of Quasipattern Solutions of the Swift-Hohenberg Equation

<p>We consider the steady Swift-Hohenberg partial differential equation, a one-parameter family of PDEs on the plane that models, for example, Rayleigh-B,nard convection. For values of the parameter near its critical value, we look for small solutions, quasiperiodic in all directions of the plane, and which are invariant under rotations of angle . We solve an unusual small divisor problem and prove the existence of solutions for small parameter values, then address their stability with respect to quasi-periodic perturbations.</p>