stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

periodic solutions of the lienard equation with one sided growth restrictions in this article we are concerned with the problem of the existence of periodic solutions to the periodically forced scalar lienard equation x f x x g x e t d dt 1 where f g e r r are continuous functions and e is periodic we are looking for solutions of 1 having the same period of the forcing term e throughout the paper we assume without loss of generality see e g 22 36 p 693 that

contributions to coincidence degree theory of asymptotically homogeneous operators in this paper we give contributions to the coincidence degree theory of asymptotically homogeneous operators applications are given to the periodic problem for second order functional differential equations

nonresonance conditions for asymptotically positively homogeneous differential systems the fucik spectrum and its generalization in this article we propose how to define the fucik spectra for differential systems in rnwith jumping nonlinearities the propertyp a generalization of the non fucik spectra to positively homogeneous nonautonomous differential systems is also defined in the most general setting a quantitative estimate result concerning the parameterized completely continuous positively homogeneous perturbations of the identity in banach spaces is given in this article such a result is a generalization of one property in the spectrum theory for completely continuous linear operators using this estimate result we will give some significantly improved nonresonance results from the fucik spectra and propertypfor asymptotically positively homogeneous differential systems

periodic solutions of second order equations via rotation numbers abstract we consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems we show how a rotation number approach together with the poincare birkhoff theorem and the phase plane analysis of the spiral properties allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity these systems may also be resonant in particular our approach can be used to deal with the problems without both global existence of the cauchy problems associated to the equation and the sign assumption on f the typical example is a partially superlinear second order equation our main result generalizes some classical results of jacobowitz and hartman among others

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

continuation theorems for periodic perturbations of autonomous systems it is first shown in this paper that whenever it exists the coincidence degree of the left hand member of an autonomous differential equation x g x 0 in the space of periodic functions with fixed period omega can be computed in terms of the brouwer degree of g this result provides efficient continuation theorems specially for omega periodic perturbations of autonomous systems extensions to differential equations in flow invariant enr s are also given

wave breaking for shallow water models with time decaying solutions abstract a family of camassa holm type equations with a linear term and cubic and quartic nonlinearities is considered local well posedness results are established via kato s approach conserved quantities for the equation are determined and from them we prove that the energy functional of the solutions is a time dependent monotonically decreasing function of time and bounded from above by the sobolev norm of the initial data under some conditions the existence of wave breaking phenomenon is investigated and necessary conditions for its existence are obtained in our framework the wave breaking is guaranteed among other conditions when the coefficient of the linear term is sufficiently small which allows us to interpret the equation as a linear perturbation of some recent camassa holm type equations considered in the literature

stability properties of steady water waves with vorticity we present two stability analyses for exact periodic traveling water waves with vorticity the first approach leads in particular to linear stability properties of water waves for which the vorticity decreases with depth the second approach leads to a formal stability property for long water waves that have small vorticity and amplitude although we do not use a small amplitude or long wave approximation 2006 wiley periodicals inc

periodic solutions of lienard equations with singular forces of repulsive type in this paper we will use the coincidence degree to give an existence result of periodic solutions for the scalar lienard equations with singular forces of repulsive type the main result of this paper clearly describes what balance conditions between the singular force at the singularity and at the infinity should be imposed if the equation has a periodic solution as an application we will improve the results of ding ye and wang for the brillouin electron beam focusing system

on the instability of elliptic traveling wave solutions of the modified camassa holm equation abstract this paper is concerned with the orbital instability for a specific class of periodic traveling wave solutions with the mean zero property and large spatial period related to the modified camassa holm equation these solutions called snoidal waves are written in terms of the jacobi elliptic functions to prove our result we use the abstract method of grillakis shatah and strauss 23 the floquet theory for periodic eigenvalue problems and the n gaps potentials theory of dubrovin matveev and novikov 19

periodic solutions of semilinear duffing equations with impulsive effects abstract in this paper we are concerned with the existence of periodic solutions for semilinear duffing equations with impulsive effects firstly for its autonomous equation any motion of the solution is same as the motion of the corresponding equation without impulses until it meets the first impulse time under the influence of impulses these two motions are likely to be quite different we introduce some reasonable assumptions on the impulsive functions to control these differences such that the information valid for the equation without impulses can always be used for the impulsive one basing on poincare birkhoff twist theorem we prove the existence of infinitely many periodic solutions secondly as for the nonautonomous equation where the autonomous case is taken as an auxiliary one we find the relation between the solutions of these two equations and then obtain the existence of infinitely many periodic solutions also by poincare birkhoff twist theorem lastly an example with special impulses satisfying the above assumptions is given

well posedness travelling waves and geometrical aspects of generalizations of the camassa holm equation abstract in this paper we consider a five parameter equation including the camassa holm and the dullin gottwald holm equations among others we prove the existence and uniqueness of solutions of the cauchy problem using kato s approach conservation laws of the equation up to second order are also investigated from these conservation laws we establish some properties for the solutions of the equation and we also find a quadrature for it the quadrature obtained is of capital importance in a classification of bounded travelling wave solutions we also find some explicit solutions given in terms of elliptic integrals finally we classify the members of the equation describing pseudo spherical surfaces

prevalence of stable periodic solutions for duffing equations abstract we study the prevalence of stable periodic solutions of the duffing equations for external force which guarantees the existence of periodic solutions both the dissipative case and the conservative case are considered the nonlinearity may be periodic or satisfies the condition of landesman lazer type

on the set of harmonic solutions of periodically perturbed autonomous differential equations on manifolds where g m r and f r m r are continuous vector fields tangent to a not necessarily closed boundaryless differentiable manifold m r with f t periodic with respect to the first variable we investigate the structure of the set of solution pairs of 1 i e of those pairs x 0 ct m such that x is a necessarily t periodic solution of 1 we give conditions ensuring the existence of a non compact connected component of solution pairs x of 1 which emanates from the set of constant solutions of the unperturbed equation

a geometric approach to periodically forced dynamical systems in presence of a separatrix we consider the periodic boundary value problem for the non autonomous scalar second order equation x f x x e t with e a continuous and t periodic forcing term using a continuation theorem adapted from capietto et al trans amer math soc 329 1992 41 72 we propose some new conditions for the existence of t periodic solutions to the forced equation in terms of the dynamical properties of the trajectories of the associated autonomous equation x f x x 0 special emphasis will be addressed to the study of the case in which the presence of an unbounded separatrix for the autonomous system in the phase plane allows to obtain a priori bounds for the t periodic solutions of the homotopic equation x f x x e t

local well posedness and stability of solitary waves for the two component dullin gottwald holm system abstract in this paper we study the cauchy problem for the two component dullin gottwald holm system with the initial data u 0 0 b p r s r b p r s 1 r 1 p r and s max 1 1 p 3 2 2 1 p which generalizes some previous local well posed results in sobolev spaces then we prove that all the smooth solitary wave solutions are orbitally stable under small disturbances

on the well posedness problem and the scattering problem for the dullin gottwald holm equation in this paper we study the well posedness of the cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation the so called dgh equation which was derived by dullin gottwald and holm the issue of passing to the limit as the dispersive parameter tends to zero for the solution of the dgh equation is investigated and the convergence of solutions to the dgh equation as 2 0 is studied and the scattering data of the scattering problem for the equation can be explicitly expressed the new exact peaked solitary wave solutions are obtained in the dgh equation after giving the condition of existing peakon in the dgh equation it turns out to be nonlinearly stable for the peakon in the dgh equation

local well posedness and stability of peakons for a generalized dullin gottwald holm equation abstract we establish the local well posedness for a generalized dullin gottwald holm equation by using kato s theory furthermore the orbital stability of the peaked solitary waves is also proved

double resonance with landesman lazer conditions for planar systems of ordinary differential equations abstract we prove the existence of periodic solutions for first order planar systems at resonance the nonlinearity is indeed allowed to interact with two positively homogeneous hamiltonians both at resonance and some kind of landesman lazer conditions are assumed at both sides we are thus able to obtain as particular cases the existence results proposed in the pioneering papers by lazer and leach 1969 27 and by frederickson and lazer 1969 18 our theorem also applies in the case of asymptotically piecewise linear systems and in particular generalizes fabry s results in fabry 1995 10 for scalar equations with double resonance with respect to the dancer fucik spectrum

periodic solutions for second order equations with time dependent potential via time map in this paper the approach via time map is used to investigate the existence of periodic solutions for second order equation with time dependent potential our theorem generalized the results obtained by ding habets omari and zanolin which are for time independent potential and relate the behavior of the primitive of the nonlinearity with respect to the fucik spectrum of the periodic problem

tetromino tilings and the tutte polynomial we consider tiling rectangles of size 4m 4n by t shaped tetrominoes each tile is assigned a weight that depends on its orientation and position on the lattice for a particular choice of the weights the generating function of tilings is shown to be the evaluation of the multivariate tutte polynomial zg q v known also to physicists as the partition function of the q state potts model on an m 1 n 1 rectangle g where the parameter q and the edge weights v can take arbitrary values depending on the tile weights

persistence properties for a generalized camassa holm equation in weighted lp spaces abstract in present paper we study the cauchy problem for a generalized camassa holm equation which was discovered by hakkaev and kirchev 2005 12 our purpose here is to establish persistence properties and some unique continuation properties of the solutions of this equation in weighted l p spaces l p l p r p d x our results extends the work of brandolese 2012 2 on persistence properties to more general equation with higher order nonlinearities

a new fixed point theorem and periodic solutions of nonconservative weakly coupled systems abstract in this paper we prove a new fixed point theorem for the coupling of twist conditions and poincare bohl type conditions as its applications we prove the existence of periodic solutions for various nonconservative mixed type weakly coupled systems

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

periodic orbits of radially symmetric keplerian like systems a topological degree approach abstract we are concerned with non autonomous radially symmetric systems with a singularity which are t periodic in time by the use of topological degree theory we prove the existence of large amplitude periodic solutions whose minimal period is an integer multiple of t precise estimates are then given in the case of keplerian like systems showing some resemblance between the orbits of those solutions and the circular orbits of the corresponding classical autonomous system

periodic solutions for ordinary differential equations with sublinear impulsive effects abstract the continuation method of topological degree is used to investigate the existence of periodic solutions for ordinary differential equations with sublinear impulsive effects the applications of the abstract approach include the generalizations of some classical nonresonance theorem for impulsive equations for instance the existence theorem for asymptotically positively homogeneous differential systems and the existence theorem for second order equations with landesman lazer conditions

global weak solutions for the dullin gottwald holm equation abstract we prove the existence and uniqueness of global weak solutions to the dullin gottwald holm equation provided the initial data satisfies certain conditions

the existence of periodic solutions for a class of nonautonomous lattices with bounded coupling abstract in this paper by using a definition of topological degree and limiting arguments we study the existence of periodic solutions of a class of one dimensional nonautonomous chains with bounded coupling an approach is developed to deal with the case of lack of the a priori bounds for an infinite lattices

coexistence states for periodic competitive kolmogorov systems abstract we prove the existence of positive periodic solutions for a class of nonautonomous competitive periodic kolmogorov systems which generalize the may leonard model a necessary and sufficient condition is also obtained

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

dominos and the gaussian free field we define a scaling limit of the height function on the domino tiling model dimer model on simply connected regions in z 2 and show that it is the massless free field a gaussian process with independent coefficients when expanded in the eigenbasis of the laplacian

periodic solutions of superlinear impulsive differential equations a geometric approach abstract a geometric method is introduced to study superlinear second order differential equations with impulsive effects basing on a reference continuous polar lifting of a planar orientation preserving homeomorphism we prove via the poincare birkhoff twist theorem the existence of infinitely many periodic solutions of conservative superlinear second order equations with the finite twist impulsive terms we also prove by developing a new twist fixed point theorem the existence of periodic solutions for non conservative superlinear second order equations with degenerate impulsive terms

positively homogeneous hamiltonian systems in the plane i try to give a general description of the dynamics of the solutions for a planar hamiltonian system with positively homogeneous hamiltonian function and periodic forcing term most of the results obtained are already known in the special case of a scalar second order differential equation with asymmetric nonlinearity

periodic motion of a system of two or three charged particles we provide necessary and sufficient conditions for the existence of t periodic solutions of a system of second order ordinary differential equations that models the motion of two or three collinear charged particles of the same sign

invariants and wave breaking analysis of a camassa holm type equation with quadratic and cubic non linearities abstract a non local evolution equation of the camassa holm type with dissipation is considered the local well posedness of the solutions of the cauchy problem involving the equation is established via kato s approach and the wave breaking scenario is also described to prove such result we firstly construct conserved currents for the equation and from them conserved quantities

exact steady periodic water waves with vorticity we consider the classical water wave problem described by the euler equations with a free surface under the influence of gravity over a flat bottom we construct two dimensional inviscid periodic traveling waves with vorticity they are symmetric waves whose profiles are monotone between each crest and trough we use bifurcation and degree theory to construct a global connected set of such solutions c 2004 wiley periodicals inc

periodic solutions for planar autonomous systems with nonsmooth periodic perturbations abstract in this paper we consider a class of planar autonomous systems having an isolated limit cycle x 0 of smallest period t 0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1 we consider two functions defined by means of the eigenfunctions of the adjoint of the linearized system and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of t periodic solutions of the autonomous system when it is perturbed by nonsmooth t periodic nonlinear terms of small amplitude we also show the convergence of these periodic solutions to x 0 as the perturbation disappears and we provide an estimation of the rate of convergence the employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach commonly employed in the existing literature on this subject based on various versions of the implicit function theorem

periodic solutions for functional differential equations of mixed type in this paper we study the existence of periodic solutions for some functional differential equations of mixed type among the main results are the averaging principle and existence theorems for some equations with homogeneous nonlinearities we use here the coincidence degree theory of mawhin

classification of bounded travelling wave solutions for the dullin gottwald holm equation abstract in this paper we classify all bounded travelling wave solutions for the integrable dullin gottwald holm equation it is shown that it decomposes in two known cases the camassa holm and the korteweg de vries equation for the former the classification is similar to the one presented in 2005 12 while for the latter it is only possible to obtain smooth solutions

tilings of rectangles with t tetrominoes we prove that any two tilings of a rectangular region by t tetrominoes are connected by moves involving only 2 and 4 tiles we also show that the number of such tilings is an evaluation of the tutte polynomial the results are extended to a more general class of regions

maslov index poincare birkhoff theorem and periodic solutions of asymptotically linear planar hamiltonian systems abstract we study the relationship between the twist condition in the poincare birkhoff fixed point theorem and the assumptions on the maslov index for asymptotically linear planar hamiltonian systems for this aim we develop a variant of the poincare birkhoff theorem which together with its classical version allows to obtain a lower bound for the number of nontrivial periodic solutions in terms of the gap between the maslov indexes associated to the linearizations of the planar system at zero and at infinity