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

fractals in surface growth with power law noise the authors present a microscopic description of interface growth with power law noise distribution in the form p eta eta 1 approximately 1 mu which exhibits non universal roughening for the mu d 1 case in d 1 dimensions the existence of a fractal pattern in the bulk of the aggregate is explained leading trivially to the proof of the identity alpha z 2 for the roughening and the dynamical scaling exponents alpha and z respectively investigations on the distribution of step sizes of the discretized interface and the saturated growth speed further support the arguments

rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

a note on well posedness for camassa holm equation in this note we investigate the problem of well posedness for a shallow water equation with data having critical regularity our results are based on the use of besov spaces b2 rs which generalize the sobolev spaces hs with critical index s 3 2

on the cauchy problem for the two component camassa holm system in this paper we establish the local well posedness for the two component camassa holm system in a range of the besov spaces we also derive a wave breaking mechanism for strong solutions in addition we determine the exact blow up rate of such solutions to the system

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

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

gradient flow and front propagation with boundary contact energy we consider a gradient flow for the functional f epsilon u int omega frac 1 epsilon w u epsilon nabla u 2 dx int partial omega sigma u d h n 1 x epsilon 0 with a view towards modelling the interfacial motion associated with domain coarsening in binary alloys in the presence of boundary contact energy here w is a non negative double well free energy density and u is the composition density of the system the function sigma represents the contact energy between the alloy and the boundary partial omega of the container the functional was first proposed by cahn in a different context as a model for the energy of a two fluid mixture which takes into account boundary contact energy at the boundary of the fluid domain we derive the first term in an asymptotic expansion as epsilon 0 for the solution of the associated gradient flow begin eqnarray partial t u 2 epsilon delta u 1 epsilon w u x in omega t geq 0 u x 0 g epsilon x x in omega nabla u cdot bf n 1 2 epsilon sigma u x in partial omega t geq 0 end eqnarray using multiple time scales we show that fronts rapidly develop and then propagate with normal velocity epsilon kappa where kappa is the mean curvature of the front at the intersection of a front with partial omega the boundary contact energy is shown to imply a contact angle condition for the front several examples are presented for this type of propagation in the plane

non uniform dependence on initial data for the two component novikov system considered herein is the initial value problem for the two component novikov system it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs hs 1 for s 5 2 on the line based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

universal tree structures in directed polymers and models of evolving populations by measuring or calculating coalescence times for several models of coalescence or evolution with and without selection we show that the ratios of these coalescence times become universal in the large size limit and we identify a few universality classes

pattern propagation in nonlinear dissipative systems abstract we discuss the problem of pattern selection in situations where a stable nonuniform state of a nonlinear dissipative system propagates into an initially unstable homogeneous region our strategy is to consider this process as a generalization of front propagation in a nonlinear diffusion problem for which rigorous results are known and we point out that these known properties are consistent with a marginal stability hypothesis that has been suggested in the theory of dendritic crystal growth we then describe a more general interpretation of the marginal stability hypothesis and finally present numerical evidence for its validity from three different pattern forming models

a fully supersymmetric akns theory we construct a fully supersymmetric bihamiltonian theory in four superfields admitting zero curvature and lax formulation this theory is an extension of the classical akns which can be recovered as a reduction other supersymmetric theories are obtained as reductions of the susy akns namely a nonlinear schrodinger a modified kdv and the manin radul kdv the susy nonlinear schrodinger hierarchy is related to the one of roelofs and kersten we determine its bihamiltonian and lax formulation finally we show that the susy kdv s mentioned before are related through a susy miura map

blow up blow up rate and decay of the solution of the weakly dissipative camassa holm equation in this paper we mainly study several problems on the weakly dissipative periodic camassa holm equation at first the local well posedness of the equation is obtained by kato s theorem a necessary and sufficient condition of the blow up of the solution and some criteria guaranteeing the blow up of the solution are established then the blow up rate of the solution is discussed moreover we prove that the equation has global solutions and these global solutions decay to zero as time goes to infinite provided the potentials associated to their initial date are of one sign

losses in the pemc boundary perfect electromagnetic conductor pemc was recently introduced to generalize both the perfect electric conductor pec and the perfect magnetic conductor pmc which are used as boundary material in this paper an extension of pemc boundary conditions is made to a more realistic model with small losses good electromagnetic conductor it is shown that the antisymmetric boundary impedance dyadic defining the ideal pemc boundary cannot alone carry any losses and a symmetric component must be added that actually dominates the lossy behavior of the boundary as an example a slab of low loss gyrotropic wave guiding material is studied and shown to represent the lossy counterpart of the pemc boundary the effect of losses is demonstrated by considering plane wave reflection from an imperfect pemc plane

global weak solutions for a two component camassa holm shallow water system abstract in this paper we prove the existence of global weak solution for an integrable two component camassa holm shallow water system provided the initial data satisfying some certain conditions

on the global existence and wave breaking criteria for the two component camassa holm system considered herein is a two component camassa holm system modeling shallow water waves moving over a linear shear flow a wave breaking criterion for strong solutions is determined in the lowest sobolev space hs s 32 by using the localization analysis in the transport equation theory moreover an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this sobolev space hs

escape from the unstable equilibrium in a random process with infinitely many interacting particles we consider a one dimensional version of the model introduced in ref 1 at each site of z there is a particle with spin 1 particles move according to the stirring process and spins change according to the glauber dynamics in the hydrodynamical limit with the stirring process suitably speeded up the local magnetic density m r is proven in ref 1 to satisfy the reaction diffusion equation 8tmt r 1 2 rmt r v mt v m 89 4 and fl 0 c and fl being determined by the parameters of the glauber dynamics in the present paper we consider an initial state with zero magnetization mo r o we then prove that at long times before taking the hydrodynamical limit the evolution departs from that predicted by and that the microscopic state becomes a nontrivial mixture of states with different magnetizations

on the weakly dissipative camassa holm degasperis procesi and novikov equations abstract we show that the weakly dissipative camassa holm degasperis procesi hunter saxton and novikov equations can be reduced to their non dissipative versions by means of an exponentially time dependent scaling hence up to a simple change of variables the non dissipative and dissipative versions of these equations are equivalent similar results hold also for the equations in the so called b family of equations as well as for the two component and versions of the above equations

on the blow up rate and the blow up set of breaking waves for a shallow water equation we consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation blow up can occur only in the form of wave breaking i e the solution remains bounded but its slope becomes unbounded in finite time a quite detailed description of the wave breaking phenomenon is given there is at least a point in general depending on time where the slope becomes infinite exactly at breaking time the precise blow up rate is established and for a large class of initial data we also determine the blow up set

lipschitz metric for the novikov equation we consider the lipschitz continuous dependence of solutions for the novikov equation with respect to the initial data in particular we construct a finsler type optimal transport metric which renders the solution map lipschitz continuous on bounded sets of h 1 mathbb r cap w 1 4 mathbb r although it is not lipschitz continuous under the natural sobolev metric from an energy law due to the finite time gradient blowup by an application of thom s transversality theorem we also prove that when the initial data is in an open dense subset of h 1 mathbb r cap w 1 4 mathbb r the solution is piecewise smooth this generic regularity result helps us extend the lipschitz continuous metric to the general weak solutions our method of constructing the metric can be used to treat other kinds of quasi linear equations provided a good knowledge about the energy concentration

green s function extraction for interfaces with impedance boundary conditions theory and experiments to obtain the response between two receivers from cross correlations of wave fields recorded at these receivers are well established the principle relies on mutually spatial and temporal uncorrelated contributions from sources on a boundary enclosing the two receivers which upon cross correlation interfere only constructively for signals traveling between the two receivers it has therefore become generally known as interferometry the theory includes situations with flow mechanical and electromagnetic field fluctuations and their mutual coupling here we present an electromagnetic theory for green s function retrieval from cross correlations that incorporates general bianisotropic media in which interfaces are present where bianisotropic impedance boundary conditions apply the derived green s function representation shows that in lossless media and for interfaces with lossless impedance boundary conditions the green s function between two receivers is obtained by cross correlating the recordings of these receivers from sources on a boundary enclosing them we show numerical examples in 2 d where proper solutions are numerically tractable and practical approximations illustrate that numerically less intensive algorithms lead to acceptable results and accurate extraction of reflections from impedance boundary surfaces because the method is data driven it is suitable for experimental green s function extraction from measured data

blow up phenomena and global existence for a two component camassa holm system with an arbitrary smooth function abstract in the paper we mainly investigate blow up phenomena and global existence of strong solutions to a two component camassa holm systems with an arbitrary smooth function h for three types of smooth functions h by using a conservation law and the sign preserving property of strong solutions we obtain two new blow up results and a new global existence result for the two component system our obtained results generalize and cover the recent results in yan et al 2015 zhang and yin 2015 2016

diffusion of directed polymers in a random environment we consider a system of random walks or directed polymers interacting weakly with an environment which is random in space and time in spatial dimensionsd 2 we establish that the behavior is diffusive with probability one the diffusion constant is not renormalized by the interaction

conditions for the existence of higher symmetries of evolutionary equations on the lattice in this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential difference equations depending only on nearest neighboring interaction these conditions are applied to prove the existence of new integrable equations belonging to this class

on a class of physically important integrable equations abstract a methodology introduced by fuchssteiner and the author is used to derive a class of physically important integrable evolution equations among these equations are integrable generalizations of the korteweg devries kdv of the modified kdv of the nonlinear schrodinger nls and of the sine gordon equations the modeling of water waves as well as general asymptotic considerations are used to illustrate the occurrence of the generalized modified kdv and nls equations respectively

well posedness and blow up phenomena for a higher order shallow water equation this paper deals with the cauchy problem for a higher order shallow water equation yt auxy buyx 0 where y 2ku i x2 ku and k 2 the local well posedness of solutions for the cauchy problem in sobolev space hs r with s 7 2 is obtained under some assumptions the existence and uniqueness of the global solutions to the equation are shown and conditions that lead to the development of singularities in finite time for the solutions are also acquired finally the weak solution for the equation is considered

conductance distributions in random resistor networks self averaging and disorder lengths the self averaging properties of the conductanceg are explored in random resistor networks rrn with a broad distribution of bond strengthsp g g 1 the rrn problem is cast in terms of simple combinations of random variables on hierarchical lattices distributions of equivalent conductances are estimated numerically on hierarchical lattices as a function of sizel and the distribution tail strength parameter for networks above the percolation threshold convergence to a gaussian basin is always the case except in the limit 0 adisorder length d is identified beyond which the system is effectively homogeneous this length scale diverges as d v is the regular percolation correlation length exponent when the microscopic distribution of conductors is exponentially wide 0 this implies that exactly the same critical behavior can be induced by geometrical disorder and by strong bond disorder with the bond occupation probabilityp we find that only lattices at the percolation threshold have renormalized probability distributions in alevy like basin at the percolation threshold the disorder length diverges at a critical tail strength c as z withz 3 2 0 1 a new exponent critical path analysis is used in a generalized form to give the macroscopic conductance in the case of lattices abovepc

diffusive description of lattice gas models we have investigated a lattice gas model consisting of repulsive particles following deterministic dynamics two versions of the model are studied in one case we consider a finite open system in which particles can leave and enter the lattice over the edge in the other case we use periodic boundary conditions in both cases the density fluctuations exhibit a 1 f power spectrum the individual particles behave asymptotically like ordinary random walkers the collective behavior of these particles shows that due to the deterministic dynamics the particles behave as if they are correlated in time we have numerically investigated the power spectrum of the density fluctuations the lifetime distribution and the spatial correlation function we discuss the appropriate langevin like diffusion equation which can reproduce our numerical findings our conclusion is that the deterministic lattice gases are described by a diffusion equation withoutany bulk noise the open lattice gas exhibits a crossover behavior as the probability for introducing particles at the edge of the system becomes small the power spectrum changes from a 1 f to a 1 f 2 spectrum the diffusive description proven to be valid for a moderate boundary drive fails altogether when the drive goes to zero

a model for temporal fluctuations of the surface width a stochastic one dimensional map a stochastic one dimensional map is introduced to model the steady state fluctuations of the surface width in far from equilibrium surface roughening the dynamics of the map and the correlations in the time sequence are investigated in particular for power law distributed noise a non trivial multi affine behaviour is observed

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

on some wave breaking for the nonlinear integrable shallow water wave equations mathematicians in this paper based on the conservation laws and the blow up scenario some new blow up phenomena are derived for the camassa holm equation and degasperis procesi equation which have attracted much attention due to their structure c 2015 elsevier ltd all rights reserved

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

some new integrable systems constructed from the bi hamiltonian systems with pure differential hamiltonian operators abstract when both hamiltonian operators of a bi hamiltonian system are pure differential operators we show that the generalized kupershmidt deformation gkd developed from the kupershmidt deformation in 10 offers an useful way to construct new integrable system starting from the bi hamiltonian system we construct some new integrable systems by means of the generalized kupershmidt deformation in the cases of harry dym hierarchy classical boussinesq hierarchy and coupled kdv hierarchy we show that the gkd of harry dym equation gkd of classical boussinesq equation and gkd of coupled kdv equation are equivalent to the new integrable rosochatius deformations of these soliton equations with self consistent sources we present the lax pair for these new systems therefore the generalized kupershmidt deformation provides a new way to construct new integrable systems from bi hamiltonian systems and also offers a new approach to obtain the rosochatius deformation of soliton equation with self consistent sources

global solutions to a special case of the generalized weakly dissipative periodic two component camassa holm system abstract we establish global well posedness to the generalized weakly dissipative periodic two component camassa holm system in some special case

minimizers and gradient flows for singularly perturbed bi stable potentials with a dirichlet condition minimizers and gradient flows are studied for the functional w u 2 u 2 d x r n 0 where u satisfies a dirichlet condition u h on here w is taken to be a double well potential with minimum value zero attained at u a and u b questions of existence and structure of minimizers for small are resolved through the identification of a limiting variational problem the so called limit a formal asymptotic solution is then constructed for the gradient flow t u 2 u 1 w9 u u x 0 g x u x t h on valid when is small using multiple timescales we show that fronts rapidly develop and then propagate with normal velocity k where k is mean curvature at the intersection of a front with the dirichlet condition is shown to imply a contact angle condition for the front this asymptotically correct evolution process represents gradient flow for the limit

on the blow up of solutions of a periodic nonlinear dispersive wave equation in compressible elastic rods the local well posedness for a periodic nonlinear dispersive wave equation in compressible elastic rods is established blow up of solutions in finite time is obtained provided their initial data satisfy certain conditions

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

periodic solutions of the camassa holm equation based on the bilinear form since the famous work of camassa and holm various analytic techniques have been used to find the exact solutions of camassa holm ch equation and many interesting results have been found by the different methods in this paper we first give an extended bilinear form of the ch equation which contains four more free parameters than those in the results proposed in parkers finding for soliton solutions thus we construct the smooth periodic solution of the ch equation in a parametric form based on the previous work of nakumura we also show that the periodic solution degenerates into the smooth solitary wave solution by limiting behavior of is the parameter appeared from the riemann theta function

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

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

the cauchy problem for a generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem of the generalized camassa holm equation using a galerkin type approximation scheme it is shown that this equation is well posed in sobolev spaces h s s 3 2 for both the periodic and the nonperiodic case in the sense of hadamard that is the data to solution map is continuous furthermore it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous the nonuniform dependence is proved using the method of approximate solutions and well posedness estimates moreover it is shown that the solution map for the generalized camassa holm equation is holder continuous in h r topology finally with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time

on the solvability of the painleve vi equation a rigorous method was introduced by fokas and zhou 1992 for studying the riemann hilbert problem associated with the painleve ii and iv equations the same methodology has been applied to the painleve i iii and v equations in this paper we will apply the same methodology to the painleve vi equation we will show that the cauchy problem for the painleve vi equation admits in general a global meromorphic solution in t furthermore the special solution which can be written in terms of a hypergeometric function is obtained via solving the special case of the riemann hilbert problem

microscopic origin of hydrodynamic equations derivation and consequences abstract we describe some recent progress in deriving autonomous hydrodynamic type equations for macroscopic variables from model stochastic microscopic dynamics of particles on a lattice the derivations also yield the microscopic fluctuations about the deterministic macroscopic evolution these grow with time to become infinite when the deterministic solution is unstable a form of microscopic pattern selection is also found

global existence and blow up phenomena for a weakly dissipative periodic 2 component camassa holm system we first establish local well posedness for a weakly dissipative periodic 2 component camassa holm system we then present two global existence results for strong solutions to the equation we finally obtain several blow up results and the blow up rate of strong solutions to the equation

probability distribution of the interface width in surface roughening analogy with a levy flight the authors present a closed form expression for the probability distribution of the height fluctuations in the zhang model of anomalous surface roughening the result which includes both the steady state behavior and the time evolution to the steady state is based on an analogy between the d 1 1 dimensional surface problem and a d 1 levy flight in the limit case of conventional ballistic deposition they obtain a gaussian distribution for the height fluctuations their results are corroborated by detailed numerical simulations

on inverse recursion operator and tri hamiltonian formulation for a kaup newell system of dnls equations an inverse of the recursion operator is computed and a clear presentation of a tri hamiltonian formulation is provided for a kaup newell system of derivative nonlinear schrodinger dnls equations therefore all kaup newell systems in the whole hierarchy of dnls equations are tri hamiltonian and have an inverse hierarchy of common commuting symmetries

a convergent finite difference scheme for the camassa holm equation with general h 1 initial data we suggest a finite dfference scheme for the camassa holm equation that can handle general h 1 initial data the form of the difference scheme is judiciously chosen to ensure that it satisfies a total energy inequality we prove that the difference scheme converges strongly in h 1 towards a dissipative weak solution of the camassa holm equation

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

an integrable semi discretization of the camassa holm equation and its determinant solution an integrable semi discretization of the camassa holm ch equation is presented the keys of its construction are bilinear forms and determinant structure of solutions of the ch equation determinant formulas of n soliton solutions of the continuous and semi discrete camassa holm equations are presented based on determinant formulas we can generate multi soliton multi cuspon and multi soliton cuspon solutions numerical computations using the integrable semi discrete camassa holm equation are performed it is shown that the integrable semi discrete camassa holm equation gives very accurate numerical results even in the cases of cuspon cuspon and soliton cuspon interactions the numerical computation for an initial value condition which is not an exact solution is also presented

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

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

transformation method for problems involving perfect electromagnetic conductor pemc structures perfect electric conductor pec and perfect magnetic conductor pmc can be generalized to perfect electromagnetic conductor pemc a medium where certain linear combinations of electromagnetic fields are required to vanish in differential form representation the corresponding medium is characterized as the simplest possible medium it is defined through a scalar admittance parameter whose zero and infinite limits yield the pmc and pec media respectively in this paper a duality transformation is found that has the property of transforming pemc to pec and an isotropic medium to itself thus problems involving pemc objects in air can be transformed to problems with pec objects in air which can be solved through traditional techniques and then transformed back several simple examples are treated to demonstrate the principle pemc has the potential of having similar applications as pmc in antenna engineering and finding structures for its realization is a challenge

peakon solutions of the novikov equation and properties of the data to solution map abstract we consider both the periodic and the non periodic cauchy problem for the novikov equation and discuss continuity results for the data to solution map in sobolev spaces in particular we show that the data to solution map is not globally uniformly continuous in sobolev spaces with exponent less than 3 2 to accomplish this we construct sequences of peakon solutions whose distance initially goes to zero but later becomes large

the multiscale expansions of difference equations in the small lattice spacing regime and a vicinity and integrability test i we propose an algorithmic procedure i to study the distance between an integrable pde and any discretization of it in the small lattice spacing regime and at the same time ii to test the asymptotic integrability properties of such discretization this method should provide in particular useful and concrete information on how good is any numerical scheme used to integrate a given integrable pde the procedure illustrated on a fairly general ten parameter family of discretizations of the nonlinear schr dinger equation consists of the following three steps i the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in following degasperis et al 1997 physica d 100 187 211 ii the application to such an expansion of the degasperis procesi dp integrability test degasperis a and procesi m 1999 asymptotic integrability symmetry and perturbation theory spt98 ed a degasperis and g gaeta singapore world scientific pp 23 37 degasperis a 2001 multiscale expansion and integrability of dispersive wave equations lectures given at the euro summer school what is integrability isaac newton institute cambridge uk 13 24 august integrability lecture notes in physics vol 767 ed a mikhailov berlin springer to test the asymptotic integrability properties of the discrete system and its distance from its continuous limit iii the use of the main output of the dp test to construct infinitely many approximate symmetries and constants of motion of the discrete system through novel and simple formulas

the modified camassa holm equation backlund transformation and nonlinear superposition formula this paper is concerned with the modified camassa holm equation and its backlund transformation with the aid of reciprocal transformation and the associated modified camassa holm equation a backlund transformation which involves both dependent and independent variables is constructed for the modified camassa holm equation the related nonlinear superposition formula is worked out and applications of both backlund transformation and nonlinear superposition formula are considered

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

asymptotic function for multigrowth surfaces using power law noise numerical simulations are used to investigate the multiaffine exponent ensuremath alpha q and multigrowth exponent ensuremath beta q of ballistic deposition growth for noise obeying a power law distribution the simulated values of ensuremath beta q are compared with the asymptotic function ensuremath beta q 1 q that is approximated from the power law behavior of the distribution of height differences over time they are in good agreement for large q the simulated ensuremath alpha q is found in the range 1 ql ensuremath alpha q l 2 q 1 this implies that large rare events tend to break the kardar parisi zhang universality scaling law at higher order q

single and multi solitary wave solutions to a class of nonlinear evolution equations abstract in this paper an effective discrimination algorithm is presented to deal with equations arising from physical problems the aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations many examples are given to illustrate the algorithm at the same time some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations such as camassa holm equation kolmogorov petrovskii piskunov equation and so on then a direct constructive method called multi auxiliary equations expansion method is described to derive the multi solitary wave solutions of nonlinear evolution equations finally a class of novel multi solitary wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov equation are given by three direct methods the algorithm proposed in this paper can be steadily applied to some other nonlinear problems

growth anomaly and its implications abstract the growth mechanism of interfaces in nature may be anomalous in the sense that the interface s roughness is not a universal scaling function this so called growth anomaly can be traced to the non existence of a continuum limit of the growth theory various aspects of the anomaly are discussed and some experimental verifications are suggested

global existence and blow up phenomena for an integrable two component camassa holm shallow water system abstract this paper is concerned with global existence and blow up phenomena for an integrable two component camassa holm shallow water system a new global existence result and several new blow up results of strong solutions to the system are presented our obtained results for the system are sharp and improve considerably earlier results

cellular automaton fluids 1 basic theory continuum equations are derived for the large scale behavior of a class of cellular automaton models for fluids the cellular automata are discrete analogues of molecular dynamics in which particles with discrete velocities populate the links of a fixed array of sites kinetic equations for microscopic particle distributions are constructed hydrodynamic equations are then derived using the chapman enskog expansion slightly modified navier stokes equations are obtained in two and three dimensions with certain lattices viscosities and other transport coefficients are calculated using the boltzmann transport equation approximation some corrections to the equations of motion for cellular automaton fluids beyond the navier stokes order are given

well posedness of a class of solutions to an integrable two component camassa holm system abstract in this paper we study a class of solutions to the cauchy problem for an integrable two component camassa holm system with the initial data u 0 0 1 h 1 r w 1 r l 2 r l r based on characteristics we study a corresponding ode and obtain a unique local solution by applying the contraction mapping principle we prove local existence and uniqueness of the solution to the camassa holm system by constructing a solution obtained from the ode and studying the regularity of the solution finally we show continuous dependence of the solution on the initial data in some weak sense

blowup and blowup rate of solutions to a weakly dissipative periodic rod equation in this paper we first establish the local well posedness for a weakly dissipative periodic rod equation we then present a precise blowup scenario and several blowup results of strong solutions to the equation we finally give the blowup rate of strong solutions to the equation when blowup occurs our blowup results for the weakly dissipative periodic rod equation improve considerably earlier results for the periodic rod equation

factorisation of energy dependent schrodinger operators miura maps and modified systems we consider the energy dependent schrodinger operator mathbb l sum limits i 0 n lambda i varepsilon i partial 2 u i which we have previously shown to be associated with multi hamiltonian structures 2 in this paper we use an unusual form of the lax approach to derive by asingle construction the time evolutions of the eigenfunctions of mathbb l the associated hamiltonian operators and the hamiltonian functionals we then generalise the well known factorisation of standard lax operators to the case of energy dependent operators the simple product of linear factors is replaced by a dependent quadratic form we thus generalise the resulting construction of miura maps and modified equations we show that for some of our systems there exists a sequence ofn such modifications therth modification possessing n r 1 hamiltonian structures