a nonlinear liouville property for the generalized kawahara equation abstract in this paper by applying the method of martel and merle 9 we prove that if the global solution of the generalized kawahara equation gkw is close to the soliton constructed by kabakouala and molinet 4 at initial time in the energy space moreover if this solution is uniformly localized then it is identically equal to a soliton all time

periodic cauchy problem for one two dimensional generalization of the benjamin ono equation in sobolev spaces of low regularity abstract in this work we prove that the initial value problem ivp associated to the two dimensional benjamin ono equation u t h u u u x 0 x y t 2 t r u x y 0 u 0 x y where h denotes the hilbert transform with respect to the variable x and is the laplacian with respect to the spatial variables x and y is locally well posed in the periodic sobolev space h s t 2 with s 7 4

global wellposedness and limit behavior for the generalized finite depth fluid equation with small data in critical besov spaces b 2 1s abstract in this paper we study the cauchy problem for the generalized finite depth fluid equation t u g x 2 u x u k 1 k 1 0 where g i f 1 coth 2 1 2 f k is an integer that is larger than 4 we obtain that it is globally wellposed if k 4 and the initial data in b 2 s k b 2 1 s k are sufficiently small where s k 1 2 2 k and s k 1 2 1 k moreover we show that its solution will converge to those of the generalized bo and kdv equations as the depth parameter and 0 respectively

a modular functor which is universal for quantum computation we show that the topological modular functor from witten chern simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor s state space a computational model based on chern simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model the chief technical advance the density of the irreducible sectors of the jones representation has topological implications which will be considered elsewhere

control and stabilization of the benjamin ono equation in l 2 mathbb t we study the control and stabilization of the benjamin ono equation in l 2 mathbb t the lowest regularity where the initial value problem is well posed this problem was already initiated in linares and rosier trans am math soc 367 4595 4626 2015 where a stronger stabilization term was used that makes the equation of parabolic type in the control zone here we employ a more natural stabilization term related to the l2 norm moreover by proving a theorem of controllability in l2 we manage to prove the global controllability in large time our analysis relies strongly on the bilinear estimates proved in molinet and pilod anal pde 5 365 395 2012 and some new extension of these estimates established here

numerical generation of periodic traveling wave solutions of some nonlinear dispersive wave systems in this paper a numerical procedure to generate approximations to periodic traveling wave profiles of some nonlinear dispersive wave systems is introduced the method is based on a suitable modification of a fixed point algorithm of petviashvili type and solves several drawbacks of some previous algorithms presented in the literature by way of illustration the method is applied to generate numerically periodic traveling waves of two problems of interest the fractional kdv type equations and the extended boussinesq system we propose a method to compute periodic traveling waves the method can be applied to some nonlinear dispersive systems it is checked in fractional kdv equations and boussinesq systems

a robust riks like path following method for strain actuated snap through phenomena in soft solids abstract in this paper a practical and robust riks like path following method is developed to automatically trace the nonlinear equilibrium path for initial finite strain induced snap through phenomena in soft solids the deformation gradient is decomposed into an elastic part and an initial strain part where the initial strain is characterized through a prescribed pattern tensor and an unknown strain factor describing the magnitude in the non linear finite element fe solution scheme an equivalent body force proportional to the increment of the strain factor is derived which varies in each iteration step due to its dependence on deformed configuration and stress both the displacement and the strain factor increments are introduced as unknowns in the linearized equation which permits to predict the critical equilibrium states before and after the snap through several examples with various configurations and strain histories are constructed including multilayer beams and film substrate systems which demonstrate the effectiveness and robustness of the algorithm to capture the strain induced snap through phenomena

uniqueness of non linear ground states for fractional laplacians in mathbb r we prove uniqueness of ground state solutions q q x 0 of the non linear equation sq q q 1 0inr s q q q 1 0 i n r where 0 s 1 and 0 4s 1 2s for s 12 s 1 2 and 0 for s 12 s 1 2 here s denotes the fractional laplacian in one dimension in particular we answer affirmatively an open question recently raised by kenig martel robbiano and we generalize by completely different techniques the specific uniqueness result obtained by amick and toland for s 12 s 1 2 and 1 in 5 for the benjamin ono equation as a technical key result in this paper we show that the associated linearized operator l s 1 1 q is non degenerate i e its kernel satisfies ker l span q this result about l proves a spectral assumption which plays a central role for the stability of solitary waves and blowup analysis for non linear dispersive pdes with fractional laplacians such as the generalized benjamin ono bo and benjamin bona mahony bbm water wave equations

multilinear estimates for periodic kdv equations and applications we prove an endpoint multilinear estimate for the xs b spaces associated to the periodic airy equation as a consequence we obtain sharp local well posedness results for periodic generalized kdv equations as well as some global well posedness results below the energy norm

dilation method and smoothing effects of solutions to the benjamin ono equation in this paper we study smoothing effects of solutions to the benjamin ono equation where h is the hilbcrt transform defined by we prove that if h 2 and x x 4 then the solution u of bo belongs to where

a kam theorem for hamiltonian partial differential equations with unbounded perturbations we establish an abstract infinite dimensional kam theorem dealing with unbounded perturbation vector field which could be applied to a large class of hamiltonian pdes containing the derivative x in the perturbation especially in this range of application lie a class of derivative nonlinear schrodinger equations with dirichlet boundary conditions and perturbed benjamin ono equation with periodic boundary conditions so kam tori and thus quasi periodic solutions are obtained for them

on the periodic intermediate long wave equation the authors discuss some properties of a physically interesting nonlinear integro differential equation with periodic boundary conditions it is the natural periodic analogue of the intermediate long wave equation and it provides a periodic analogue of the benjamin ono equation in the appropriate limit due to the speciality of the integral operator the equation admits a backlund transformation an infinity of motion constants etc two simple periodic solutions are exhibited the equation may be transformed into more than one kind of bilinear equation

peak solutions for the fractional nirenberg problem abstract we consider the fractional nirenberg problem u k u n 2 n 2 u 0 in r n assume k x 1 e k x and k x has at least two critical points satisfying some local conditions the existence of two peak solutions is obtained when the positive number e is small enough

on uniqueness properties of solutions of the k generalized kdv equations in this paper we study uniqueness properties of solutions of the so called k generalized korteweg de vries equations our goal is to obtain sufficient conditions on the behavior of the difference u1 u2 of two solutions u1 u2 of 1 1 at two different times t0 0 and t1 1 which guarantee that u1 u2

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

some computational aspects of the stability analysis of nonlinear structures abstract the perturbation method and the continuation method are the two most popular techniques for the solution of finite element equations that describe instability phenomena this paper presents a summary of the principles involved and describes some trends in the development of these techniques

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

final state problem for class of nonlinear nonlocal dispersive equation abstract we study the large time asymptotics of the solutions to nonlinear dispersive equations of the form 0 1 t u p i x x u x u 3 t 0 x r where p i x is a pseudo differential operator defined by the symbol p in this paper we consider the case that p is nonlocal and nonhomogeneous and show the existence of the solution to 0 1 around a given modified free solution

global well posedness in l 2 for the periodic benjamin ono equation we prove that the benjamin ono equation is globally well posed in h s bbb t for s ge 0 moreover we show that the associated flow map is lipschitz on every bounded set of h s 0 bbb t s ge 0 and even real analytic in this space for small times this result is sharp in the sense that the flow map if it can be defined and coincides with the standard flow map on h 0 infty bbb t cannot be of class c 1 alpha alpha 0 from h 0 s bbb t into h 0 s bbb t as soon as s 0

on the stability of the solitary waves to the generalized kawahara equation in this paper we investigate the orbital stability of solitary waves to the generalized kawahara equation gkw which is a fifth order dispersive equation for some values of the power of the nonlinearity we prove the orbital stability in the energy space h 2 r of two branches of even solitary waves of gkw by combining the well known spectral method introduced by benjamin 3 with continuity arguments we construct the first family of even solitons by applying the implicit function theorem in the neighborhood of the explicit solitons of gkw found by dey et al 8 the second family consists of even travelling waves with low speeds they are solutions of a constraint minimization problem on the line and rescaling of perturbations of the soliton of gkdv with speed 1

coherent structures in nonlocal dispersive active dissipative systems we analyze coherent structures in nonlocal dispersive active dissipative nonlinear systems using as a prototype the kuramoto sivashinsky ks equation with an additional nonlo cal term that contains stabilizing destabilizing and dispersive parts as for the local generalized kuramoto sivashinsky gks equation see e g t kawahara and s toh phys fluids 31 1988 pp 2103 2111 we show that sufficiently strong dispersion regularizes the chaotic dynamics of the ks equation and the solutions evolve into arrays of interacting pulses that can form bound states we analyze the asymptotic characteristics of such pulses and show that their tails tend to zero al gebraically but not exponentially as for the local gks equation since the shilnikov type approach is not applicable for analyzing bound states in nonlocal equations we develop a weak interaction theory and show that the standard first neighbor approximation is no longer applicable it is then essential to take into account long range interactions due to the algebraic decay of the tails of the pulses in addition we find that the number of possible bound states for fixed parameter values is always finite and we determine when there is long range attractive or repulsive force between the pulses finally we explain the regularizing effect of dispersion by showing that as dispersion is increased the pulses generally undergo a transition from absolute to convective instability we also find that for some nonlocal operators increasing the strength of the stabilizing destabilizing term can have a regularizing deregularizing effect on the dynamics

a new zk ilw equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism abstract the research of algebraic gravity solitary waves is an advanced field which has important practical and theoretical value in physical oceanography aerology and etc by calculation condition and theoretical method limit previous researches mainly focused on the 1 1 dimensional models and 2 1 dimensional models were few considered in this paper from the non static equilibrium equation a new zk ilw equation is derived by using multi scale analysis and perturbation method in finite depth stratified atmosphere which is the first time obtained the model can reduce to zk bo model h and zk model h 0 and is the generalization of the above two models in order to further understand the nature of algebraic gravity solitary waves we get the analytical solution of zk ilw equation by using the trial function method and discuss the conservation laws furthermore the fission process of algebraic gravity solitary waves is studied and we can judge that one of the possible formation mechanism of squall lines is the fission of algebraic gravity solitary waves

an application of the inverse scattering transform to the modified intermediate long wave equation the modified intermediate long wave milw equation is a 1 1 dimensional nonlinear singular integro differential equation that possesses soliton solutions in an appropriate limit the milw equation reduces to the well known modified korteweg de vries equation in this paper we solve the initial value problem for the milw equation through a suitable implementation of the inverse scattering transform and use of the miura type transformation that maps solutions of the milw equation into solutions of a complexified version of the standard intermediate long wave ilw equation the initial value used for the milw equation is assumed to be real valued sufficiently smooth and decaying to zero as the absolute value of the spatial variable approaches large values an interesting feature of the procedure we develop is that soliton solutions for the ilw and milw equations can be derived by appropriate specializations of a master set of equations

blow up boundary solutions of some semilinear fractional equations in the unit ball for 0 we are interested in blow up solutions u c b of the fractional problem in the unit ball b we distinguish particularly two orders of singularity at the boundary solutions exploding at the same rate than 2 1 denotes the euclidean distance and those higher singular than 2 1 as a consequence it will be shown that the classical keller osserman condition cannot be readopted in the fractional setting

operator splitting for the benjamin ono equation abstract in this paper we analyze operator splitting for the benjamin ono equation u t u u x h u x x where h denotes the hilbert transform if the initial data are sufficiently regular we show the convergence of both godunov and strang splittings

existence of minimizers of functionals involving the fractional gradient in the absence of compactness symmetry and monotonicity abstract we establish general assumptions under which a constrained variational problem involving the fractional gradient and a local nonlinearity admits minimizers

benjamin ono equation with unbounded data abstract the initial value problem ivp for the benjamin ono equation with unbounded initial data is considered we show existence and uniqueness of global solutions for sublinear growth data the method of proof relies in an appropriate splitting the study of an ivp associated with a benjamin ono type equation with variable coefficients and the persistence of decay for solutions of the burgers equation

sufficient conditions for orbital stability of periodic traveling waves abstract the present paper deals with sufficient conditions for orbital stability of periodic waves of a general class of evolution equations supporting nonlinear dispersive waves our method can be seen as an extension to spatially periodic waves of the theory of solitary waves recently developed in 25 firstly our main result do not depend on the parametrization of the periodic wave itself secondly motived by the well known orbital stability criterion for solitary waves we show that the same criterion holds for periodic waves in addition we show that the positiveness of the principal entries of the hessian matrix related to the energy surface function are also sufficient to obtain the stability consequently we establish the orbital stability of periodic waves for several nonlinear dispersive models we believe our method can be applied in a wide class of evolution equations in particular it can be extended to regularized dispersive wave equations

the cauchy problem for dissipative benjamin ono equation in weighted sobolev spaces abstract we study the well posedness in weighted sobolev spaces for the initial value problem ivp associated with the dissipative benjamin ono dbo equation we establish persistence properties of the solution flow in the weighted sobolev spaces z s r h s r l 2 x 2 r d x s r 0 we also prove some unique continuation properties in these spaces in particular such results of unique continuation show that our results of well posedness are sharp

local well posedness for the nonlocal derivative nonlinear schrodinger equation in besov spaces abstract in this paper we study the cauchy problem associated with the one dimensional integro differential nonlocal derivative nonlinear schrodinger equation in the besov space b 2 1 2 1 r the local well posedness for small initial data in b 2 1 2 1 r is established our method of proof combines the contraction principle applied to the associated integral equation together with interpolations of some smoothing effects kato s smoothing effects strichartz estimate and estimates for the maximal function for phase localized functions associated to the linear dispersive part of the equation and a fractional vector valued leibniz s rule derived by molinet and ribaud in 2004

direct and inverse scattering problems of the nonlinear intermediate long wave equation the inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed the equation describes long internal gravity waves in a stratified fluid of finite depth and reduces to the korteweg de vries equation as shallow water limit and the benjamin ono equation as deep water limit both limits of the method and novel aspects of the theory are also discussed

unique continuation for some evolution equations this work is concerned with unique continuation results for some evolution equations essentially of parabolic type to be more precise let l be an evolution operator acting on functions defined on some connected open set b of r r x r l is said to have the unique continuation property if every solution u of lu 0 which vanishes one some nonempty open set u i of fl vanishes in the horizontal component of o in 0 i e in x t e 0 3x x t e co the first general result of this type was established by mizohata 6 in the case where l is a second order parabolic operator with smooth coefficients however mizohata s proof is somewhat delicate it is based in particular on an extended class of calderon zygmund operators and does not lead obviously to a weakening of the smoothness of the coefficients on the other hand it is important for the applications to deal with nonsmooth coefficients since l is often obtained as the linearization of some nonlinear operator at a nonnecessarily smooth solution we refer to lions 4 and kernevez and lions 3 for applications to control theory and to 1 l for applications to transversality techniques let us now describe the content of this work in the first section we prove a unique continuation theorem when l is a second order parabolic equation our proof is simple and based on the derivation of a carleman estimate which is reminiscent of the classical carleman estimates for second order elliptic operators see 121 this carleman inequality allows the weakening of the 118 0022 0396187 3 00

uniqueness properties of solutions to the benjamin ono equation and related models abstract we prove that if u 1 u 2 are real solutions of the benjamin ono equation defined in x t r 0 t which agree in an open set r 0 t then u 1 u 2 we extend this uniqueness result to a general class of equations of benjamin ono type in both the initial value problem and the initial periodic boundary value problem this class of 1 dimensional non local models includes the intermediate long wave equation we relate our uniqueness results with those for a water wave problem finally we present a slightly stronger version of our uniqueness results for the benjamin ono equation

statistical mechanics of the periodic benjamin ono equation the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations

control and stabilization of the benjamin ono equation on a periodic domain it was proved by linares and ortega in 24 that the linearized benjamin ono equation posed on a periodic domain t with a distributed control supported on an arbitrary subdomain is exactly controllable and exponentially stabilizable the aim of this paper is to extend those results to the full benjamin ono equation a feedback law in the form of a localized damping is incorporated in the equation a smoothing effect established with the aid of a propagation of regularity property is used to prove the semi global stabilization in l 2 t of weak solutions obtained by the method of vanishing viscosity the local well posedness and the local exponential stability in h s t are also established for s 1 2 by using the contraction mapping theorem finally the local exact controllability is derived in h s t for s 1 2 by combining the above feedback law with some open loop control

well posedness for a higher order benjamin ono equation in this paper we prove that the initial value problem associated to the following higher order benjamin ono equation tv bh x2v a x3v cv xv d x vh xv h v xv where x t r v is a real valued function h is the hilbert transform a r b c and d are positive constants is locally well posed for initial data v 0 v0 hs r s 2orv0 hk r l2 r x2dx k z k 2

on the limit from the intermediate long wave equation to the benjamin ono equation the intermediate long wave equation is a physically important singular integrodifferential equation containing a parameter referred to here as for it reduces to the benjamin ono equation it has been recently shown that the inverse scattering transform schemes of the above equations have certain significant differences here it is shown that for the inverse scattering transform scheme of the intermediate long wave equation reduces to that of the benjamin ono equation

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

long time behavior for the equation of finite depth fluids in this paper we study the cauchy problem for the generalized equation of finite depth fluids partial t u g partial x 2 u partial x left frac u p p right 0 whereg is a singular integral andp is an integer larger than 1 we obtain the long time behavior of the fundamental solution of linear problem and prove that the solutions of the nonlinear problem with small initial data for p 5 2 sqrt 21 2 are decay in time and freely asymptotic to solutions of the linear problem in addition we also study some properties of the singular integralg inl q r withq 1

uniqueness of radial solutions for the fractional laplacian we prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional laplacian s with s 0 1 for any space dimensions n 1 by extending a monotonicity formula found by cabre and sire we show that the linear equation s u vu 0 in r n has at most one radial and bounded solution vanishing at infinity provided that the potential v is radial and nondecreasing in particular this result implies that all radial eigenvalues of the corresponding fractional schrodinger operator h s v are simple furthermore by combining these findings on linear equations with topological bounds for a related problem on the upper half space r n 1 we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation s q q q q 0 in r n for arbitrary space dimensions n 1 and all admissible exponents 0 this generalizes the nondegeneracy and uniqueness result for dimension n 1 recently obtained by the first two authors and in particular the uniqueness result for solitary waves of the benjamin ono equation found by amick and toland

weakly damped forced korteweg de vries equations behave as a finite dimensional dynamical system in the long time abstract it is shown that a weak dissipation of order zero is sufficient to impose a finite dimensional behavior for korteweg de vries equations with periodic boundary conditions a universal attractor which captures all the trajectories is constructed its fractal dimension is proved to be finite

smoothing properties and retarded estimates for some dispersive evolution equations smoothing properties in the form of space time integrability properties play an important role in the study of dispersive evolution equations a number of them follow from a combination of general arguments and specific estimates we present a general formulation which makes the separation between the two types of ingredients as clear as possible and we illustrate it with the examples of the schrodinger equation of the wave equation and of a class of 1 1 dimensional equations related to the benjamin ono equation of special interest for the cauchy problem are retarded estimates expressed in terms of those properties we derive a number of such estimates associated with the last example and we mention briefly an application of those estimates to the cauchy problem for the generalized benjamin ono equation

decoherence free subspaces of anyon states abstract in this paper we propose decoherence free subspaces dfss of anyon states under two kinds of noises the forms of collective dephasing and collective rotation noises are introduced the invariance requirement under these noises give some restrictions of the winding number of anyons we put forward a scheme to realize the dfs of anyon states in the kitaev spin lattice model and give some discussion of general kinds of anyons

the cauchy problem and stability of solitary wave solutions for rlw kp type equations abstract the kadomtsev petviashvilli kp equation ut ux uux uxxx x euyy 0 arises in various contexts where nonlinear dispersive waves propagate principally along the x axis but with weak dispersive effects being felt in the direction parallel to the y axis perpendicular to the main direction of propagation we propose and analyze here a class of evolution equations of the form ut ux upux lut x euyy 0 which provides an alternative to eq in the same way the regularized long wave equation is related to the classical korteweg de vries kdv equation the operator l is a pseudo differential operator in the x variable p 1 is an integer and e 1 after discussing the underlying motivation for the class a local well posedness theory for the initial value problem is developed with assumptions on l and p that include conditions appertaining to models of interesting physical phenomenon the solutions defined locally in time t are shown to be smoothly extendable to the entire time axis in the particularly interesting case where l x2 and e 1 possesses travelling wave solutions u x y t c x ct y provided c 1 and 0 0 4 3 and c 1 and for 4 3 if c 4p 4 p the paper concludes with commentary on extensions of the present theory to more than two space dimensions

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

stability of nonlinear periodic internal waves in a deep stratified fluid stability of the nonlinear periodic solutions of the benjamin ono equation with respect to one dimensional perturbations and taking into account transverse modulations is analytically studied the full set of eigenfrequencies and of small perturbations is found in the one dimensional case the dispersion equation for long wave oblique perturbations is obtained these perturbations are shown to be unstable under either sign of the dispersion of the linear dispersion relation the instability limits and the maximum growth rates are determined in the extreme cases of strongly and weakly nonlinear waves

refined energy inequality with application to well posedness for the fourth order nonlinear schrodinger type equation on torus we consider the time local and global well posedness for the fourth order nonlinear schrodinger type equation 4nls on the torus the nonlinear term of 4nls contains the derivatives of unknown function and this prevents us to apply the classical energy method to overcome this difficulty we introduce the modified energy and derive an a priori estimate for the solution to 4nls

stability of solitary waves for the vector nonlinear schrodinger equation in higher order sobolev spaces abstract in this manuscript a sharp form of orbital stability for the schrodinger system also known as the vector nls i t u j 2 x x u j 2 i 1 m u i 2 u j 0 where u j are complex valued functions of x t r 2 j 1 2 m is established in l 2 based sobolev classes of arbitrarily high order the result means practically that not only does the bulk of what emanates from the perturbed solitary wave stay close in shape and propagation and phase speeds to the original solitary wave but emerging residual oscillations must also be very small and not only in the energy norm

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation

on the generalized benjamin ono equation we study well posedness of the initial value problem for the generalized benjamin ono equation tu ukdxu 0 k e in sobolev spaces hs r for small data and higher nonlinearities k 2 new local and global including scattering results are established our method of proof is quite general it combines several estimates concerning the associated linear problem with the contraction principle hence it applies to other dispersive models in particular it allows us to extend the results for the generalized benjamin ono to nonlinear schrodinger equations or systems of the form atu i2u p u oxu ui axi 0

smoothing properties and existence of solutions for the generalized benjamin ono equation abstract we study the cauchy problem for the generalized benjamin ono equation tu d d 2 u dv u where d d dx u 0 and v e 1 r r with v 0 v 0 0 using commutator identities and estimates we prove that the equation exhibits smoothing properties similar to those of the generalized korteweg de vries equation gkdv which is the special case 1 of if v satisfies suitable estimates at zero and at infinity we prove that the cauchy problem for the equation with initial data u 0 u0 has a solution u l r l2 l2loc r h loc if u0 l2 and has a solution u l r h l2loc r h2 loc if u0 h finite energy solution in addition we prove that the usual benjamin ono equation 1 2 v u u 2 has a solution u l r h1 l2loc r h 3 2 loc for u0 h1 those results are easily extended to related equations such as the intermediate long wave equation or the smith equation

high order predictor corrector algorithms new predictor corrector algorithms are presented for the computation of solution paths of non linear partial differential equations the predictors and the correctors are based on perturbation techniques and pade approximants this extends the asymptotic numerical method anm which is an efficient high order continuation technique without corrector the efficiency and the reliability of the new technique are assessed by several examples within thin shell theory and navier stokes equations many variants have been tested to establish an optimal algorithm

well posedness in energy space for the periodic modified benjamin ono equation abstract we prove that the periodic modified benjamin ono equation is locally well posed in the energy space h 1 2 this ensures the global well posedness in the defocusing case the proof is based on an x s b analysis of the system after a gauge transform

on the stochastic benjamin ono equation abstract we discuss the cauchy problem for the stochastic benjamin ono equation in the function class h s r s 3 2 when there is a zero order dissipation we also establish the existence of an invariant measure with support in h 2 r many authors have discussed the cauchy problem for the deterministic benjamin ono equation but our results are new for the stochastic benjamin ono equation our goal is to extend known results for the deterministic equation to the stochastic equation

sufficient conditions for stability of solitary wave solutions of model equations for long waves abstract we consider solitary wave solutions of model equations for long waves that feature a general form of linear dispersion sufficient conditions for the non linear stability of such solutions are derived these conditions are shown to obtain for the korteweg de vries equation and certain of its generalizations such as the benjamin ono equation and the intermediate long wave equation

a numerical scheme for periodic travelling wave simulations in some nonlinear dispersive wave models a numerical method for simulating periodic travelling wave solutions of some nonlinear dispersive wave equations is proposed the construction of the scheme is based on an efficient computation of the elements that characterize these solutions the initial profile and the velocity of the wave

large time asymptotics of solutions to the generalized benjamin ono equation we study the asymptotic behavior for large time of solutions to the cauchy problem for the generalized benjamin ono bo equation ut u 1u x huxx 0 where h is the hilbert transform x t r when the initial data are small enough if the power of the nonlinearity is greater than 3 then the solution of the cauchy problem has a quasilinear asymptotic behavior for large time in the case 3 critical for the asymptotic behavior i e for the modified benjamin ono equation we prove that the solutions have the same l time decay as in the corresponding linear bo equation also we find the asymptotics for large time of the solutions of the cauchy problem for the bo equation in the critical and noncritical cases for the critical case we prove the existence of modified scattering states if the initial function is sufficiently small in certain weighted sobolev spaces these modified scattering states differ from the free scattering states by a rapidly oscillating factor

a nonlinear finite element model for the stress analysis of soft solids with a growing mass abstract the paper presents a new finite element fe model for the stress analysis of soft solids with a growing mass based on the work of lubarda and hoger 2002 contrary to the traditional numerical methods emphasizing on the influence of growth on constitutive equations an equivalent body force is firstly detected which is resulted from the linearization of the nonlinear equation and acts as the driver for material growth in the numerical aspect in the algorithm only minor correction on the traditional tangent modulus is needed to take the growth effects into consideration and its objectivity could be guaranteed comparing with the traditional method to solve the resulted equation in time domain both explicit and implicit integration algorithms are developed where the growth tensor is updated as an internal variable of gauss point the explicit updating scheme shows higher efficiency while the implicit one seems to be more robust and accurate the algorithm validation and its good performance are demonstrated by several two dimensional examples including free growth constrained growth and stress dependent growth

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

why should anyone care about computing with anyons in this article we present a pedagogical introduction of the main ideas and recent advances in the area of topological quantum computation we give an overview of the concept of anyons and their exotic statistics present various models that exhibit topological behaviour and establish their relation to quantum computation possible directions for the physical realization of topological systems and the detection of anyonic behaviour are elaborated

on the propagation of regularities in solutions of the benjamin ono equation abstract we shall deduce some special regularity properties of solutions to the ivp associated to the benjamin ono equation mainly for datum u 0 h 3 2 r whose restriction belongs to h m b for some m z m 2 and b r we shall prove that the restriction of the corresponding solution u t belongs to h m for any r and any t 0 therefore this type of regularity of the datum travels with infinite speed to its left as time evolves

the ivp for the benjamin ono equation in weighted sobolev spaces abstract in this work we continue our study initiated in fonseca and ponce 2011 11 on the uniqueness properties of real solutions to the ivp associated to the benjamin ono bo equation in particular we shall show that the uniqueness results established in fonseca and ponce 2011 11 do not extend to any pair of non vanishing solutions of the bo equation also we shall prove that the uniqueness result established in fonseca and ponce 2011 11 under a hypothesis involving information of the solution at three different times cannot be relaxed to two different times

nonlinear hamiltonian waves with constant frequency and surface waves on vorticity discontinuities waves with constant nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves we propose an inviscid burgers hilbert equation as a model equation for such waves and give a dimensional argument to show that it models hamiltonian surface waves with constant frequency using the method of multiple scales we derive a cubically nonlinear quasi linear nonlocal asymptotic equation for weakly nonlinear solutions we show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two dimensional inviscid incompressible fluid flows thus the burgers hilbert equation provides an effective equation for these waves we describe the hamiltonian structure of the burgers hilbert and asymptotic equations and show that the asymptotic equation can also be derived by means of a near identity transformation we derive a semiclassical approximation of the asymptotic equation and show that spatially periodic harmonic traveling waves are linearly and modulationally stable numerical solutions of the burgers hilbert and asymptotic equations are in excellent agreement in the appropriate regime in particular the lifespan of small amplitude smooth solutions of the burgers hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation 2009 wiley periodicals inc

global well posedness in the energy space for the benjamin ono equation on the circle we prove that the benjamin ono equation is locally well posed in h 1 2 mathbb t this leads to a global well posedeness result in h 1 2 mathbb t thanks to the energy conservation

generalized internal long waves equations construction hamiltonian structure and conservation laws a general class of the ilw type equations is constructed we introduce a hamiltonian structure and construct an infinite number of conservation laws

sharp ill posedness result for the periodic benjamin ono equation abstract we prove the discontinuity for the weak l 2 t topology of the flow map associated with the periodic benjamin ono equation this ensures that this equation is ill posed in h s t as soon as s 0 and thus completes exactly the well posedness result obtained in molinet 2008 12

orbital stability of periodic traveling wave solutions for the kawahara equation in this paper we investigate the orbital stability of periodic traveling waves for the kawahara equation we prove that the periodic traveling wave under certain conditions minimizes a convenient functional by using an adaptation of the method developed by grillakis et al j funct anal 74 160 197 1987 the required spectral properties to ensure the orbital stability are obtained by knowing the positiveness of the fourier transform of the associated periodic wave established by angulo and natali siam j math anal 40 1123 1151 2008

an arc length method including line searches and accelerations this paper describes a method for introducing line searches into the arc length solution procedure such line searches may be used at each iteration to calculate an optimum scalar step length which scales the normal iterative vector in practice a loose tolerance is provided so that on many iterations the line searches are avoided however on difficult iterations the line searches are shown to lead to a substantial improvement in the convergence characteristics a simple single parameter acceleration is also developed using line search concepts the new arc length method is applied to both the geometrically nonlinear analysis of shallow shells and the materially nonlinear analysis of reinforced concrete beams and slabs significant improvements are demonstrated in relation to the standard arc length method

construction of a minimal mass blow up solution of the modified benjamin ono equation we construct a minimal mass blow up solution of the modified benjamin ono equation mbo begin aligned u t u 3 d 1 u x 0 end aligned mbo which is a standard mass critical dispersive model let q in h frac 1 2 q 0 be the unique ground state solution of d 1 q q q 3 constructed using variational arguments by weinstein commun part differ equations 12 1133 1173 1987a j differ equations 69 192 203 1987b and albert et al proc r soc lond a 453 1233 1260 1997 and whose uniqueness was recently proved by frank and lenzmann acta math 210 261 318 2013 we show the existence of a solution s of mbo satisfying vert s vert l 2 vert q vert l 2 and begin aligned s t frac 1 lambda frac 1 2 t q left frac cdot x t lambda t right rightarrow 0 quad text in h frac 1 2 mathbb r text as t downarrow 0 end aligned where begin aligned lambda t sim t quad x t sim ln t quad hbox and quad vert s t vert dot h frac 1 2 sim t frac 1 2 vert q vert dot h frac 1 2 quad hbox as t downarrow 0 end aligned this existence result is analogous to the one obtained by martel et al j eur math soc 17 1855 1925 2015 for the mass critical generalized korteweg de vries equation however in contrast with the gkdv equation for which the blow up problem is now well understood in a neighborhood of the ground state s is the first example of blow up solution for mbo the proof involves the construction of a blow up profile energy estimates as well as refined localization arguments developed in the context of benjamin ono type equations by kenig et al ann inst h poincare anal non lin 28 853 887 2011 due to the lack of information on the mbo flow around the ground state the energy estimates have to be considerably sharpened in the present paper

dynamics of a nonlocal kuramoto sivashinsky equation abstract in this paper we study the effects of a nonlocal term on the global dynamics of the kuramoto sivashinsky equation we show that the equation possesses a family of maximal attractors parameterized by the mean value of the initial data the dimension of the attractor is estimated as a function of the coefficient of the nonlocal term and the width of the periodic domain

existence and finite dimensionality of the global attractor for a class of nonlinear dissipative equations in this work we discuss the cauchy problem for a class of nonlinear dissipative equations as well as the existence of a global attractor and we estimate its dimension in the sense of the hausdorff or fractal dimension

on the mechanics of solids with a growing mass a general constitutive theory of the stress modulated growth of biomaterials is presented with a particular accent given to pseudo elastic soft living tissues the governing equations of the mechanics of solids with a growing mass are revisited within the framework of finite deformation continuum thermodynamics the multiplicative decomposition of the deformation gradient into its elastic and growth parts is employed to study the growth of isotropic transversely isotropic and orthotropic biomaterials an explicit representation of the growth part of the deformation gradient is given in each case which leads to an effective incremental formulation in the analysis of the stress modulated growth process the rectangular components of the instantaneous elastic moduli tensor are derived corresponding to selected forms of the elastic strain energy function physically appealing structures of the stress dependent evolution equations for the growth induced stretch ratios are proposed

on well posedness for the benjamin ono equation we prove existence and uniqueness of solutions for the benjamin ono equation with data in h s mathbb r s 1 4 moreover the flow is holder continuous in weaker topologies