stability in the energy space of the sum of n peakons for the degasperis procesi equation the degasperis procesi equation possesses well known peaked solitary waves that are called peakons their stability has been established by lin and liu in 5 in this paper we localize the proof in some suitable sense detailed in section 3 of the stability of a single peakon thanks to this we extend the result of stability to the sum of n peakons traveling to the right with respective speeds c1 cn such that the difference between consecutive locations of peakons is large enough

stability of two soliton collision for nonintegrable gkdv equations we continue our study of the collision of two solitons for the subcritical generalized kdv equations label kdvabs partial t u partial x partial x 2 u f u 0 quad quad 0 1 solitons are solutions of the type u t x q c 0 x x 0 c 0 t where c0 0 in 21 mainly devoted to the case f u u4 we have introduced a new framework to understand the collision of two solitons q c 1 q c 2 for 0 1 in the case c 2 ll c 1 or equivalently q c 2 h 1 ll q c 1 h 1 in this paper we consider the case of a general nonlinearity f u for which q c 1 q c 2 are nonlinearly stable in particular since f is general and c1 can be large the results are not perturbations of the ones for the power case in 21

stability in the energy space of the sum of n peakons for a modified camassa holm equation with higher order nonlinearity considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons

size and shape analysis of landmark data summary the paper deals with statistical analysis of landmark data using the exact joint distribution theory for the size and shape of planar gaussian configurations this is an extension of earlier work which considered marginal shape analysis special cases of the size and shape distribution are examined and the isotropic gaussian model is investigated in particular detail various properties are studied including conditional distributions and a curve of regression finally we consider some possible approaches to inference including exact maximum likelihood estimation of size and shape a practical biological application is given which investigates the size and shape of mouse vertebrae

stability and asymptotic stability in the energy space of the sum of n solitons for subcritical gkdv equations we prove in this paper the stability and asymptotic stability in h1 of a decoupled sum of n solitons for the subcritical generalized kdv equations the proof of the stability result is based on energy arguments and monotonicity of the local l2 norm note that the result is new even for p 2 the kdv equation the asymptotic stability result then follows directly from a rigidity theorem in 16

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

stability of the train of n solitary waves for the two component camassa holm shallow water system abstract considered herein is the integrable two component camassa holm shallow water system derived in the context of shallow water theory which admits blow up solutions and the solitary waves interacting like solitons using modulation theory and combining the almost monotonicity of a local version of energy with the argument on the stability of a single solitary wave we prove that the train of n solitary waves which are sufficiently decoupled is orbitally stable in the energy space h 1 r l 2 r

statistical modeling of curves using shapes and related features motivated by the problems of analyzing protein backbones diffusion tensor magnetic resonance imaging dt mri fiber tracts in the human brain and other problems involving curves in this study we present some statistical models of parameterized curves in in terms of combinations of features such as shape location scale and orientation for each combination of interest we identify a representation manifold endow it with a riemannian metric and outline tools for computing sample statistics on these manifolds an important characteristic of the chosen representations is that the ensuing comparison and modeling of curves is invariant to how the curves are parameterized the nuisance variables including parameterization are removed by forming quotient spaces under appropriate group actions in the case of shape analysis the resulting spaces are quotient spaces of hilbert spheres and we derive certain wrapped truncated normal densities for capturing variability in observed curves we demonstrate

description of two soliton collision for the quartic gkdv equation in this paper we give the rst description of the collision of two solitons for a nonintegrable equation in a special regime we consider solutions of the quartic gkdv equation tu x 2 xu u 4 0 which behave as t 1 like u t x qc1 x c1t qc2 x c2t t x where qc x ct is a soliton andk t kh1k qc2kh1k qc1kh1 the global behavior of u t is given by the following stability result for all t 2 r u t x qc1 t x y1 t qc2 t x y2 t t x where k t kh1k qc2kh1 and limt 1c1 t c limt 1c2 t c in the case where u t is a pure 2 soliton solution as t 1 i e limt 1 k t kh1 0 we obtain c c1 c 0 therefore in contrast with the integrable kdv equation or mkdv equation no global pure 2 soliton solution exists and the collision is inelastic a dierent notion of global 2 soliton is then proposed

spectral stability and time evolution of n solitons in the kdv hierarchy this paper concerns spectral stability and time evolution of n solitons in the korteweg de vries kdv hierarchy with mixed commuting time flows spectral stability problem is analysed by using a pair of self adjoint operators with finite numbers of negative eigenvalues we show that the absence of unstable eigenvalues in the stability problem is related to the absence of negative eigenvalues of these operators in the constrained function spaces time evolution of n solitons is uniquely characterized from the inverse scattering transform technique

studies on finite sturmian words several properties of finite sturmian words are proved the inverse of the richomme seebold bijection between factor sets of given length of sturmian sequences and left special sturmian words is constructed using circular factors of christoffel words and studied the labbe bijection between christoffel words and left special sturmian words is introduced and studied factor sets are classified by two types the two classes are characterized by conjugation and periodicity properties the two classes are related to the rauzy graph a normal form for finite sturmian words is given following a result of de luca and de luca several classes of sturmian words are characterized through this normal form left or right special words bispecial words palindromes central words the normal form is used to derive a proof of the lipatov mignosi formula for the number of sturmian words of given length to produce a linear algorithm which checks if a word is sturmian and which computes its normal form to construct algorithmically the contraction and completion in various classes of sturmian words to compute the de luca palindromic closure of any sturmian word and prove that it is sturmian the completion is related to the de luca palindromization function and it is shown that the sets of directive words of central words of even and odd length are rational group languages christoffel words are characterized in several ways an extension of a result of chuan an extension of a result of droubay pirillo by counting circular factors by counting lyndon factors a special christoffel word is constructed having length a given markoff number a bivariate count of special sturmian words is given using ideas of bedaride domenjoud jamet and remy open problems are given

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

asymptotic stability of high dimensional zakharov kuznetsov solitons we prove that solitons or solitary waves of the zakharov kuznetsov zk equation a physically relevant high dimensional generalization of the korteweg de vries kdv equation appearing in plasma physics and having mixed kdv and nonlinear schrodinger nls dynamics are strongly asymptotically stable in the energy space we also prove that the sum of well arranged solitons is stable in the same space orbital stability of zk solitons is well known since the work of de bouard proc r soc edinburgh 126 89 112 1996 our proofs follow the ideas of martel siam j math anal 157 759 781 2006 and martel and merle math ann 341 391 427 2008 applied for generalized kdv equations in one dimension in particular we extend to the high dimensional case several monotonicity properties for suitable half portions of mass and energy we also prove a new liouville type property that characterizes zk solitons and a key virial identity for the linear and nonlinear part of the zk dynamics obtained independently of the mixed kdv nls dynamics this last virial identity relies on a simple sign condition which is numerically tested for the two and three dimensional cases with no additional spectral assumptions required possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well posedness theory in the energy space and the verification of a corresponding sign condition

construction of solutions to the subcritical gkdv equations with a given asymptotical behavior we consider the subcritical generalized korteweg de vries equation ut uxx u4 x 0 t x r let rj t x qcj x xj cjt j 1 n be n soliton solutions to this equation denote u t the kdv linear group and let v be in an adequate weighted sobolev space we construct a solution u t to the generalized korteweg de vries equation such that limt u t u t v j 1nrj t h1 0

stability in h 1 of the sum of k solitary waves for some nonlinear schrodinger equations in this article we consider nonlinear schrodinger nls equations in rd for d 1 2 and 3 we consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable let rk t x be k solitary wave solutions of the equation with different speeds v1 v2 vk provided that the relative speeds of the solitary waves vk vk 1 are large enough and that no interaction of two solitary waves takes place for positive time we prove that the sum of the rk t is stable for t 0 in some suitable sense in h1 to prove this result we use an energy method and a new monotonicity property on quantities related to momentum for solutions of the nonlinear schrodinger equation this property is similar to the l2 monotonicity property that has been proved by martel and merle for the generalized korteweg de vries gkdv equations see 12 lem 16 proof of prop 6 and that was used to prove the stability of the sum of k solitons of the gkdv equations by the authors of the present article see 15 th 1 i

character theory of monoids over an arbitrary field abstract the basic character theory of finite monoids over the complex numbers was developed in the sixties and seventies based on work of munn ponizovski mcalister rhodes and zalcstein in particular mcalister determined the space of functions spanned by the irreducible characters of a finite monoid over c and the ring of virtual characters in this paper we present the corresponding results over an arbitrary field as a consequence we obtain a quick proof of the theorem of berstel and reutenauer that the characteristic function of a regular cyclic language is a virtual character of the free monoid this is a crucial ingredient in their proof of the rationality of the zeta function of a sofic shift in symbolic dynamics

asymptotic stability of the toda m soliton we prove that multi soliton solutions of the toda lattice are both linearly and nonlinearly stable our proof uses neither the inverse spectral method nor the lax pair of the model but

asymptotic stability of solitons for subcritical generalized kdv equations we prove in this paper the asymptotic completeness of the family of solitons in the energy space for generalized korteweg de vries equations in the subcritical case this includes in particular the kdv equation and the modified kdv equation this result is obtained as a consequence of a rigidity theorem on the flow close to a soliton up to a scaling and a translation which has its own interest the proofs use some tools introduced in a previous paper to prove similar results in the case of critical generalized kdv equation

asymptotic stability of solitary waves we show that the family of solitary waves 1 solitons of the korteweg de vries equation partial t u u partial x u partial x 3 u 0 is asymptotically stable our methods also apply for the solitary waves of a class of generalized korteweg de vries equations partial t u partial x f u partial x 3 u 0 in particular we study the case wheref u up 1 p 1 p 1 2 3 and 3 0 withf c4 the same asymptotic stability result for kdv is also proved for the casep 2 the modified korteweg de vries equation we also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 the solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep 4 the solution is decomposed into a modulating solitary wave with time varying speedc t and phase t bound state part and an infinite dimensional perturbation radiating part the perturbation is shown to decay exponentially in time in a local sense relative to a frame moving with the solitary wave asp 4 the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

stability of periodic peakons for the modified μ camassa holm equation abstract considered herein is the dynamical stability of periodic peaked solitons for the modified camassa holm equation with the cubic nonlinearity the equation is a version of the modified camassa holm equation and is integrable with the lax pair and bi hamiltonian structure the equation admits the periodic peakons it is shown that the periodic peakons are dynamically stable under small perturbations in the energy space by finding inequalities related to the first three conservation laws with global maximum and minimum of the solution

minimal automation for a factorial transitive and rational language abstract we prove that every ftr language r factorial transitive rational admits a minimal irreducible deterministic automaton this automaton can be computed in three different ways first it admits an intrinsic definition depending only on the syntactic monoid of the language r secondly it is a quotient for a simple equivalence relation of every irreducible deterministic automaton recognizing r thirdly it is a subautomaton of the classical minimal deterministic automaton associated with r computable with a simple algorithm

asymptotic stability of solitons of the gkdv equations with general nonlinearity we consider the generalized korteweg de vries equation gkdv partial t u partial x partial x 2 u f u 0 quad t x in 0 t times mathbb r with general c3 nonlinearity f under an explicit condition on f and c 0 there exists a solution in the energy space h1 of the type u t x qc x x0 ct called soliton in this paper under general assumptions on f and qc we prove that the family of solitons around qc is asymptotically stable in some local sense in h1 i e if u t is close to qc for all t 0 then u t locally converges in the energy space to some qc as t note in particular that we do not assume the stability of qc this result is based on a rigidity property of the gkdv equation around qc in the energy space whose proof relies on the introduction of a dual problem these results extend the main results in martel siam j math anal 38 759 781 2006 martel and merle j math pures appl 79 339 425 2000 arch ration mech anal 157 219 254 2001 nonlinearity 1 55 80 devoted to the pure power case

asymptotic stability of solitons for mkdv abstract we prove a full asymptotic stability result for solitary wave solutions of the mkdv equation we consider small perturbations of solitary waves with polynomial decay at infinity and prove that solutions of the cauchy problem evolving from such data tend uniformly on the real line to another solitary wave as time goes to infinity we describe precisely the asymptotics of the perturbation behind the solitary wave showing that it satisfies a nonlinearly modified scattering behavior this latter part of our result relies on a precise study of the asymptotic behavior of small solutions of the mkdv equation

description of the inelastic collision of two solitary waves for the bbm equation we prove that the collision of two solitary waves of the bbm equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space we show precise estimates of the nonzero residue due to the collision moreover we give a precise description of the collision phenomenon change of size of the solitary waves and shifts in their trajectories to prove these results we extend the method introduced in martel and merle description of two soliton collision for the quartic gkdv equation submitted preprint http arxiv org abs 0709 2672 commun math phys 286 39 79 2009 for the generalized kdv equation in particular in the quartic case the main argument is the construction of an explicit approximate solution in the collision region

orbital stability of double solitons for the benjamin ono equation in this paper we prove the orbital stability of double solitons for the benjamin ono equation in the case of the kdv equation this stability has been proved in 17 parts of the proof given there rely on the fact that the euler lagrange equations for the conserved quantities of the kdv equation are ordinary differential equations since this is not the case for the benjamin ono equation new methods are required our approach consists in using a new invariant for multi solitons and certain new identities motivated by the sylvester law of inertia

sofic dyck shifts we define the class of sofic dyck shifts which extends the class of markov dyck shifts introduced by inoue krieger and matsumoto sofic dyck shifts are shifts of sequences whose finite factors form unambiguous context free languages we show that they correspond exactly to the class of shifts of sequences whose sets of factors are visibly pushdown languages we give an expression of the zeta function of a sofic dyck shift

on the orbital stability of the degasperis procesi antipeakon peakon profile in this paper we prove an orbital stability result for the degasperis procesi peakon with respect to perturbations having a momentum density that is first negative and then positive this leads to the orbital stability of the antipeakon peakon profile with respect to such perturbations

inelasticity of soliton collisions for the 5d energy critical wave equation for the focusing energy critical wave equation in 5d we construct a solution showing the inelastic nature of the collision of two solitons for any choice of sign speed scaling and translation parameters except the special case of two solitons of same scaling and opposite signs beyond its own interest as one of the first rigorous studies of the collision of solitons for a non integrable model the case of the quartic gkdv equation being partially treated in martel and merle ann math 174 2 757 857 2011 invent math 183 3 563 648 2011 int math res notices 2015 3 688 739 2015 this result can be seen as part of a wider program aiming at establishing the soliton resolution conjecture for the critical wave equation this conjecture has already been proved in the 3d radial case in duyckaerts et al camb j math 1 75 144 2013 and in the general case in 3 4 and 5d along a sequence of times in duyckaerts et al geom funct anal 27 4 798 862 2017 compared with the construction of an asymptotic two soliton in martel and merle arch ration mech anal 222 3 1113 1160 2016 the study of the nature of the collision requires a more refined approximate solution of the two soliton problem and a precise determination of its space asymptotics to prove inelasticity these asymptotics are combined with the method of channels of energy from duyckaerts et al camb j math 1 75 144 2013 kenig et al geom funct anal 24 610 647 2014

orbital stability of the train of peakons for an integrable modified camassa holm equation abstract in this paper we consider the modified camassa holm equation with cubic nonlinearity which is integrable and admits the single peakons and multi peakons using energy argument and combining the method of the orbital stability of a single peakon with monotonicity of the local energy norm we prove that the sum of n sufficiently decoupled peakons is orbitally stable in the energy space

stability of the sum of two solitary waves for gdnls in the energy space abstract in this paper we continue the study in 18 we use the perturbation argument modulational analysis and the energy argument in 15 16 to show the stability of the sum of two solitary waves with weak interactions for the generalized derivative schrodinger equation gdnls in the energy space here gdnls hasn t the galilean transformation invariance the pseudo conformal invariance and the gauge transformation invariance and the case 1 we considered corresponds to the l 2 supercritical case

shifts of k nested sequences we introduce a new class of subshifts of sequences called k graph shifts which expresses nested constraints on k symbols instead of on two symbols like for dyck shifts these shifts share many properties with markov dyck shifts but are generally not conjugate to them we prove that they are conjugate to sofic dyck shifts we give a computation of the multivariate zeta function for this class of shifts

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

existence of blow up solutions in the energy space for the critical generalized kdv equation from these conservation laws h appears as an energy space so that it is a natural space in which to study the solutions note that p 2 is a special case for equation 2 indeed from the integrability theory see lax 14 we have for suitable u0 u0 and its derivatives with fast decay at infinity an infinite number of conservation laws the general question is to understand the dynamics induced by such equations local existence in time of solutions of 2 in the energy space is now well understood see kato 10 ginibre and tsutsumi 8 for the h theory s 32 kenig ponce and vega 11 for the l theory in the case of equation 1 and sharp h theory for 2 and bourgain 3 and 4 for the periodic case

asymptotic stability of peakons for the novikov equation abstract the novikov equation is an integrable camassa holm type equation with a cubic nonlinearity one of the most important features of this equation is the existence of peaked traveling waves also called peakons this paper aims to prove the asymptotic stability of peakon solutions under h 1 r perturbations satisfying that their associated momentum density defines a non negative radon measure motivated by molinet s work 24 25 26 we shall first prove a liouville property for h 1 r global solutions belonging to a certain class of almost localized functions more precisely we show that such solutions have to be a peakon the main novelty in our analysis in comparison to the camassa holm equation comes from the fact that in our present case the momentum is not a conserved quantity and may be unbounded along the trajectory in this regard to prove the liouville property we used a new lyapunov functional not related to the not conserved momentum of the equation

equation of motion for interacting pulses we develop a systematic method of deriving the equation of motion for interacting fronts or pulses in one dimension the theory is applicable to both dissipative and dispersive systems in the case of the time dependent ginzburg landau equation which is a typical example of a dissipative system the front equation obtained is the same as has been obtained previously the pulse interaction is also derived for the korteweg char21 de vries equation emphasizing the difference between the cases with and without dissipative terms

stability of blow up profile and lower bounds for blow up rate for the critical generalized kdv equation the generalized korteweg de vries equations are a class of hamiltonian systems in infinite dimension derived from the kdv equation where the quadratic term is replaced by a higher order power term these equations have two conservation laws in the energy space h 1 l 2 norm and energy we consider in this paper the critical generalized kdv equation which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in h 1 uniform in time for all h 1 solutions and thus global existence from 15 there do exist for this equation solutions u t such that u t h1 as t t where t we call them blow up solutions the question is to describe in a qualitative way how blow up occurs for solutions with l 2 mass close to the minimal mass allowing blow up and with decay in l 2 at the right we prove after rescaling and translation which leave invariant the l 2 norm that the solution converges to a universal profile locally in space at the blow up time t from the nature of this profile we improve the standard lower bound on the blow up rate for finite time blow up solutions

asymptotic n soliton like solutions of the subcritical and critical generalized korteweg de vries equations we consider the generalized korteweg de vries equations ut uxx upx 0t x r inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in the subcritical and critical cases p 2 3 4 or 5 let r j t x qc j x c j t x j where j 1 n be n soliton solutions of this equation with corresponding speeds 0 c 1 c 2 c n in this paper we construct a solution u t of the generalized korteweg de vries equation such that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i this solution behaves asymptotically as t as the sum of n solitons without loss of mass by dispersion this is an exceptional behavior indeed being given the parameters c j 1 j n x j 1 j n we prove uniqueness of such a solution in the integrable cases p 2 and 3 such solutions are explicitly known and their properties were extensively studied in the literature they are called n soliton solutions therefore the existence result is new only for the nonintegrable cases the uniqueness result is new for all cases

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

orbital stability of periodic peakons to a generalized μ camassa holm equation in this paper we study the orbital stability of the periodic peaked solitons of the generalized camassa holm equation with nonlocal cubic and quadratic nonlinearities the equation is a version of a linear combination of the camassa holm equation and the modified camassa holm equation it is also integrable with the lax pair and bi hamiltonian structure and admits the single peakons and multi peakons by constructing an inequality related to the maximum and minimum of solutions with the conservation laws we prove that even in the case that the camassa holm energy counteracts in part the modified camassa holm energy the shapes of periodic peakons are still orbitally stable under small perturbations in the energy space

effective dynamics of double solitons for perturbed mkdv we consider the perturbed mkdv equation partial t u partial x partial x 2u 2u 3 b x t u where the potential b x t b 0 hx ht 0 h ll 1 is slowly varying with a double soliton initial data on a dynamically interesting time scale the solution is mathcal o h 2 close in h2 to a double soliton whose position and scale parameters follow an effective dynamics a simple system of ordinary differential equations these equations are formally obtained as hamilton s equations for the restriction of the mkdv hamiltonian to the submanifold of solitons the interplay between algebraic aspects of complete integrability of the unperturbed equation and the analytic ideas related to soliton stability is central in the proof

nonlinear stability of gardner breathers abstract we show that breather solutions of the gardner equation a natural generalization of the kdv and mkdv equations are h 2 r stable through a variational approach we characterize gardner breathers as minimizers of a new lyapunov functional and we study the associated spectral problem through i the analysis of the spectrum of explicit linear systems spectral stability and i i controlling degenerated directions by using low regularity conservation laws

the gardner equation and the l 2 stability of the n soliton solution of the korteweg de vries equation multi soliton solutions of the korteweg de vries equation kdv are shown to be globally l2 stable and asymptotically stable in the sense of martel and merle the proof is surprisingly simple and combines the gardner transform which links the gardner and kdv equations together with the martel merle tsai and martel merle recent results on stability and asymptotic stability in the energy space applied this time to the gardner equation as a by product the results of maddocks sachs and merle vega are improved in several directions

construction of solutions to the l 2 critical kdv equation with a given asymptotic behaviour we consider the critical korteweg de vries equation ut uxx u x 0 t x r let rj t x qcj x xj cjt j 1 n be n soliton solutions to this equation denote u t the kdv linear group and let v h be with sufficient decay on the right that is 1 x0 v l for some 0 0 we construct a solution u t to the critical korteweg de vries equation such that lim t u t u t v n x j 1 rj t h1 0

zeta functions of formal languages motivated by symbolic dynamics and algebraic geometry over finite fields we define cyclic languages and the zeta function of a language the main result is that the zeta function of a cyclic language which is recognizable by a finite automation is rational

asymptotic stability of n solitary waves of the fpu lattices we study stability of n solitary wave solutions of the fermi pasta ulam fpu lattice equation solitary wave solutions of the fpu lattice equation cannot be characterized as critical points of conservation laws due to the lack of infinitesimal invariance in the spatial variable in place of standard variational arguments for hamiltonian systems we use an exponential stability property of the linearized fpu equation in a weighted space which is biased in the direction of motion the dispersion of the linearized fpu equation balances the potential term for low frequencies whereas the dispersion is superior for high frequencies we approximate the low frequency part of a solution of the linearized fpu equation by a solution to the linearized korteweg de vries kdv equation around an n soliton solution we prove an exponential stability property of the linearized kdv equation around n solitons by using the linearized backlund transformation and use the result to analyze the linearized fpu equation

refined jacobian estimates and gross pitaevsky vortex dynamics we study the dynamics of vortices in solutions of the gross pitaevsky equation i u t delta u 1 over varepsilon 2 u left 1 u 2 right in a bounded simply connected domain omega subset mathbb r 2 with natural boundary conditions on previous rigorous results have shown that for sequences of solutions u varepsilon with suitable well prepared initial data one can determine limiting vortex trajectories and moreover that these trajectories satisfy the classical ode for point vortices in an ideal incompressible fluid we prove that the same motion law holds for a small but fixed varepsilon and we give estimates of the rate of convergence and the time interval for which the result remains valid the refined jacobian estimates mentioned in the title relate the jacobian j u of an arbitrary function u in h 1 omega mathbb c to its ginzburg landau energy in the analysis of the gross pitaevsky equation they allow us to use the jacobian to locate vortices with great precision and they also provide a sort of dynamic stability of the set of multi vortex configurations

the orbital stability of the ground states and the singularity formation for the gravitational vlasov poisson system we study the gravitational vlasov poisson system f t v cdot nabla x f e cdot nabla v f 0 0 where e x nabla x phi x delta x phi rho x rho x int mathbb r n f x v rm d x rm d v in dimension n 3 4 in dimension n 3 where the problem is subcritical we prove using concentration compactness techniques that every minimizing sequence to a large class of minimization problems attained on steady states solutions are up to a translation shift relatively compact in the energy space this implies in particular the orbital stability in the energy space of the spherically symmetric polytropes and improves the nonlinear stability results obtained for this class in 11 16 19 in dimension n 4 where the problem is l1 critical we obtain the polytropic steady states as best constant minimizers of a suitable sobolev type inequality relating the kinetic and the potential energy we then derive using an explicit pseudo conformal symmetry the existence of critical mass finite time blow up solutions and prove more generally a mass concentration phenomenon for finite time blow up solutions this is the first result of description of a singularity formation in a vlasov setting the global structure of the problem is reminiscent of the one for the focusing nonlinear schrodinger equation iu t u u p 1u in the energy space h 1 mathbb r n

stability of peakons for an integrable modified camassa holm equation with cubic nonlinearity considered herein is the dynamical stability of the single peaked soliton and periodic peaked soliton for an integrable modified camassa holm equation with cubic nonlinearity the equation is known to admit a single peaked soliton and multi peakon solutions and is shown here to possess a periodic peaked soliton by constructing certain lyapunov functionals it is demonstrated that the shapes of these waves are stable under small perturbations in the energy space

inelastic interaction of nearly equal solitons for the quartic gkdv equation this paper describes the interaction of two solitons with nearly equal speeds for the quartic gkdv equation partial tu partial x partial x 2u u 4 0 quad t x in mathbb r 0 1 we call soliton a solution of 0 1 of the form u t x qc x ct y0 where c 0 y0 and q c q c 4 cq c since 0 1 is not an integrable model the general question of the collision of two given solitons q c 1 x c 1 t q c 2 x c 2 t with c1 c2 is an open problem