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

the periodic cauchy problem for a two component non isospectral cubic camassa holm system abstract in this paper we study the periodic cauchy problem for a two component non isospectral cubic camassa holm system which includes the fokas olver rosenau qiao forq or modified camassa holm mch equation and the two component mch system as two special cases the system is integrable in the sense of possessing a non isospectral lax pair with the spectrum depending on time t and admits multi peakon solutions in an explicit form furthermore we establish the local well posedness for the system in the besov space b 2 r s t with s 3 2 1 r where the key ingredients include the friedrichs regularization method the littlewood paley decomposition theory and the transport theory in besov spaces then we derive a precise blow up criteria which is dependent of the parameters t and t moreover by the intrinsic structure of the system we obtain a new blow up result for strong solutions with sufficient conditions on the initial data and parameters the entire proof procedure relies upon a newly derived transport equation which is involved in nonlocal velocity term along the characteristic curves

the supersymmetric camassa holm equation and geodesic flow on the superconformal group we study a family of fermionic extensions of the camassa holm equation within this family we identify three interesting classes a equations which are inherently hamiltonian describing geodesic flow with respect to an h1 metric on the group of superconformal transformations in two dimensions b equations which are hamiltonian with respect to a different hamiltonian structure and c supersymmetric equations classes a and b have no intersection but the intersection of classes a and c gives a system with interesting integrability properties we demonstrate the painleve property for some simple but nontrivial reductions of this system and also discuss peakon type solutions

local and global well posedness of the fractional order epdiff equation on r d abstract of concern is the study of fractional order sobolev type metrics on the group of h diffeomorphism of r d and on its sobolev completions d q r d it is shown that the h s sobolev metric induces a strong and smooth riemannian metric on the banach manifolds d s r d for s 1 d 2 as a consequence a global well posedness result of the corresponding geodesic equations both on the banach manifold d s r d and on the smooth regular frechet lie group of all h diffeomorphisms is obtained in addition a local existence result for the geodesic equation for metrics of order 1 2 s 1 d 2 is derived

on the cauchy problem for the integrable modified camassa holm equation with cubic nonlinearity abstract considered in this paper is the modified camassa holm equation with cubic nonlinearity which is integrable and admits the single peaked solitons and multi peakon solutions the short wave limit of this equation is known as the short pulse equation the main investigation is the cauchy problem of the modified camassa holm equation with qualitative properties of its solutions it is firstly shown that the equation is locally well posed in a range of the besov spaces the blow up scenario and the lower bound of the maximal time of existence are then determined a blow up mechanism for solutions with certain initial profiles is described in detail and nonexistence of the smooth traveling wave solutions is also demonstrated

persistence properties for a family of nonlinear partial differential equations abstract we examine the persistence of decay properties for a family of dispersive nonlinear partial differential equations we show that certain decay properties of the initial data persist for as long as the solution exists on the other hand for a subset of the family certain decay rates are possible only for the trivial solution for example the only solution that remains with compact support for any further time is the trivial solution

the camassa holm equation as a geodesic flow on the diffeomorphism group misiolek j geom phys 24 203 208 1998 has shown that the camassa holm equation is a geodesic flow on the bott virasoro group in this paper it is shown that the camassa holm equation for the case 0 is the geodesic spray of the weak riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the h1 inner product over the entire group this paper uses the right trivialization technique to rigorously verify that the euler poincare theory for lie groups can be applied to diffeomorphism groups the observation made in this paper has led to physically meaningful generalizations of the ch equation to higher dimensional manifolds

symplectic connections and the linearisation of hamiltonian systems this paper uses symplectic connections to give a hamiltonian structure to the first variation equation for a hamiltonian system along a given dynamic solution this structure generalises that at an equilibrium solution obtained by restricting the symplectic structure to that point and using the quadratic form associated with the second variation of the hamiltonian plus casimir as energy this structure is different from the well known and elementary tangent space construction our results are applied to systems with symmetry and to lie poisson systems in particular

constructing an atlas for the diffeomorphism group of a compact manifold with boundary with application to the analysis of image registrations this paper considers the problem of defining a parameterization chart on the group of diffeomorphisms with compact support motivated primarily by a problem in image registration where diffeomorphic warps are used to align images constructing a chart on the diffeomorphism group will enable the quantitative analysis of these warps to discover the normal and abnormal variation of structures in a population we construct a chart for particular choices of boundary conditions on the space on which the diffeomorphism acts and for a particular class of metrics on the diffeomorphism group which define a class of diffeomorphic interpolating splines the geodesic equation is computed for this class of metrics and we show how it can be solved in the spline representation furthermore we demonstrate that the spline representation generates submanifolds of the diffeomorphism group and we study this mapping explicit computational examples are included showing how this chart can be constructed in practice and that the use of the geodesic distance allows better classification of variation than those obtained using just a euclidean metric on the space of warps

gevrey solvability and gevrey regularity in differential complexes associated to locally integrable structures in this work we study some properties of the differential complex associated to a locally integrable involutive structure acting on forms with gevrey coefficients among other results we prove that for such complexes gevrey solvability follows from smooth solvability under the sole assumption of a regularity condition as a consequence we obtain the proof of the gevrey solvability for a first order linear pde with real analytic coefficients satisfying the nirenberg treves condition p

a new integrable two component system with cubic nonlinearity in this paper a new integrable two component system mt m uxvx uv uvx uxv x nt n uxvx uv uvx uxv x where m u uxx and n v vxx is proposed our system is a generalized version of the integrable system mt m ux2 u2 x which was shown having cusped solution cuspon and w m shape soliton solutions by qiao j math phys 47 112701 2006 the new system is proven integrable not only in the sense of lax pair but also in the sense of geometry namely it describes pseudospherical surfaces accordingly infinitely many conservation laws are derived through recursion relations furthermore exact solutions such as cuspons and w m shape solitons are also obtained

global well posedness for the averaged euler equations in two dimensions abstract we prove global well posedness of the two dimensional averaged euler or euler equations for initial potential vorticity of class l2 this model generalizes the one dimensional fokas fuchssteiner camassa holm equation which describes the propagation of unidirectional waves on the surface of shallow water as such it can be realized as a geodesic equation for the h1 metric on the lie algebra of vector fields moreover in two dimensions the model obeys an advection equation for the so called potential vorticity in close analogy to the vorticity form of the euler equations we construct solutions to the weak form of the potential vorticity equation by taking the inviscid limit of solutions to a system regularized by artificial viscosity since the streamfunction vorticity relation is of order four we can show uniqueness even for potential vorticities in l2

non uniform dependence and well posedness for the rotation camassa holm equation on the torus abstract in this paper we study the periodic cauchy problem for a mathematical model of the equatorial water waves propagating mainly in one direction with the weak coriolis effect due to the earth s rotation which reduces to the camassa holm equation as the coriolis effect vanishes we first prove that this model is well posed in the sense of hadamard when initial data belongs to the sobolev space h s t with s 3 2 then we show that the well posedness is sharp in the sense that the continuity of the data to solution map is not better than continuous whose proof rests upon the method of approximate solutions and the well posedness estimate however if h s t is equipped with a weaker h r topology for 0 r s we demonstrate that the data to solution map is holder continuous with the exponent depending on r and s finally we establish a cauchy kowalevski type theorem for this model

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

geometry and curvature of diffeomorphism groups withh1metric and mean hydrodynamics holm marsden and ratiu adv in math 137 1998 1 81 derived a new model for the mean motion of an ideal fluid in euclidean space given by the equationv t u t v t 2 u t t u t grad p t where div u 0 andv 1 2 u in this model the momentumvis transported by the velocityu with the effect that nonlinear interaction between modes corresponding to length scales smaller than is negligible we generalize this equation to the setting of ann dimensional compact riemannian manifold the resulting equation is the euler poincare equation associated with the geodesic flow of theh1right invariant metric on ds the group of volume preserving hilbert diffeomorphisms of classhs we prove that the geodesic spray is continuously differentiable fromtds m intottds m so that a standard picard iteration argument proves existence and uniqueness on a finite time interval our goal in this paper is to establish the foundations for lagrangian stability analysis following arnold ann inst grenoble16 1966 319 361 to do so we use submanifold geometry and prove that the weak curvature tensor of the right invarianth1metric on ds is a bounded trilinear map in thehstopology from which it follows that solutions to jacobi s equation exist using such solutions we are able to study the infinitesimal stability behavior of geodesics

a generalization of borel s theorem and microlocal gevrey regularity in involutive structures in this paper we use borel s procedure to construct gevrey approximate solutions of an initial value problem for involutive systems of gevrey complex vector fields as an application we describe the gevrey wave front set of the boundary values of approximate solutions in wedges w of gevrey involutive structures m v we prove that the gevrey wave front set of the boundary value is contained in the polar of a certain cone t w contained in rv tx where x is a maximally real edge of w we also prove a partial converse

a note on multi dimensional camassa holm type systems on the torus we present a 2n component nonlinear evolutionary pde which includes the n dimensional versions of the camassa holm and the hunter saxton systems as well as their partially averaged variations our goal is to apply arnold s geometric formalism arnold 1966 ann inst fourier grenoble 16 319 61 ebin and marsden 1970 ann math 92 102 63 to this general equation in order to obtain results on well posedness conservation laws or stability of its solutions following the line of arguments of kohlmann 2011 j phys a math theor 44 465205 we present geometric aspects of a two dimensional periodic b equation on the diffeomorphism group of the torus in this context

well posedness of the epdiff equation with a pseudo differential inertia operator abstract in this article we study the class of right invariant fractional order sobolev type metrics on groups of diffeomorphisms of a compact manifold m our main result concerns well posedness properties for the corresponding euler arnold equations also called the epdiff equations which are of importance in mathematical physics and in the field of shape analysis and template registration depending on the order of the metric we will prove both local and global well posedness results for these equations as a result of our analysis we will also obtain new commutator estimates for elliptic pseudo differential operators

well posedness and blow up solution for a new coupled camassa holm equations with peakons a new camassa holm system with two component admitting peakon solitons is proposed the local well posedness for the system is established a criterion and a condition on the initial data guaranteeing the development of singularities in finite time for strong solutions are obtained and an existence result for a class of local weak solution is also given

on microlocal analyticity and smoothness of solutions of first order nonlinear pdes we study the microlocal analyticity and smoothness of solutions u of of the nonlinear pde ut f x t u ux under some assumptions on the repeated brackets of the linearized operator and its conjugate

vanishing distance phenomena and the geometric approach to sqg in this article we study the induced geodesic distance of fractional order sobolev metrics on the groups of volume preserving diffeomorphisms and symplectomorphisms the interest in these geometries is fueled by the observation that they allow for a geometric interpretation for prominent partial differential equations in the field of fluid dynamics these include in particular the modified constantin lax majda and surface quasi geostrophic equations the main result of this article shows that both of these equations stem from a riemannian metric with vanishing geodesic distance

analyticity of the cauchy problem for an integrable evolution equation in this paper we consider the periodic cauchy problem for the camassa holm equation with analytic initial data and prove that its solutions are analytic in both variables globally in space and locally in time

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

on the blow up structure for the generalized periodic camassa holm and degasperis procesi equations considered herein are the generalized camassa holm and degasperis procesi equations in the spatially periodic setting the precise blow up scenarios of strong solutions are derived for both of equations several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established the exact blow up rates are also determined finally geometric descriptions of these two integrable equations from non stretching invariant curve flows in centro equiaffine geometries pseudo spherical surfaces and affine surfaces are given

well posedness wave breaking and peakons for a modified μ camassa holm equation considered herein is a modified periodic camassa holm equation with cubic nonlinearity which is called the modified camassa holm equation the proposed equation is shown to be formally integrable with the lax pair and bi hamiltonian structure local well posedness of the initial value problem to the modified camassa holm equation in the besov space is established existence of peaked traveling wave solutions and formation of singularities of solutions for the equation are then investigated it is shown that the equation admits a single peaked soliton and multi peakon solutions with a similar character of the camassa holm equation singularities of the solutions can occur only in the form of wave breaking and several wave breaking mechanisms for solutions with certain initial profiles are described in detail

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

microglobal regularity and the global wavefront set in this paper we begin the study of regularity of partial differential equations in the space of global l q gevrey functions recently introduced in adwan et al j geom anal 27 3 1874 1913 2017 and hoepfner and raich indiana univ math j forthcoming and in a generalized and new function space called the space of global l q denjoy carleman functions we develop a wedge approach similar to bony s theorem bony in seminaire goulaouic schwartz 1976 1977 equations aux derivees partielles et analyse fonctionnelle exp no 3 centre math ecole polytech palaiseau 1977 and prove three main theorems the first establishes the existence of boundary values of continuous functions on a wedge next we borrow the fbi transform approach from hoepfner and raich forthcoming to define global wavefront sets and prove a relationship between the inclusion of a direction in the global wavefront set and the existence of boundary values of sums of weighted l p functions defined in wedges the final result is an application in which we prove a global version of a classical result namely the relationship between the global characteristic set of a partial differential operator p and the microglobal wavefront sets of u and pu

approximate solutions and micro regularity in the denjoy carleman classes abstract we begin with a sequence m of positive real numbers and we consider the denjoy carleman class c m we show how to construct m approximate solutions for complex vector fields with c m coefficients we then use our construction to study micro local properties of boundary values of approximate solutions in general m involutive structures of codimension one where the approximate solution is defined in a wedge whose edge where the boundary value exists is a maximally real submanifold we also obtain a c m version of the edge of the wedge theorem

the degasperis procesi equation as a non metric euler equation in this paper we present a geometric interpretation of the degasperis procesi dp equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle we also show that for any evolution in the family of b equations there is neither gain nor loss of the spatial regularity of solutions this in turn allows us to view the dp and the camassa holm equation as an ode on the frechet space of all smooth functions on the circle

blow up solutions and peakons to a generalized μ camassa holm integrable equation considered here is a generalized type integrable equation which can be regarded as a generalization to both the camassa holm and modified camassa holm equations it is shown that the proposed equation is formally integrable with the lax pair and the bi hamiltonian structure and its scale limit is an integrable model of hydrodynamical systems describing short capillary gravity waves local well posedness of the cauchy problem in the suitable sobolev space is established by the viscosity method existence of peaked traveling wave solutions and formation of singularities of solutions for the equation are investigated it is found that the equation admits single and multi peaked traveling wave solutions the effects of varying camassa holm and modified camassa holm nonlocal nonlinearities on blow up criteria and wave breaking are illustrated in detail our analysis relies on the method of characteristics and conserved quantities and is proceeded with a priori differential estimates

existence of weak solutions in lower order sobolev space for a camassa holm type equation a generalized camassa holm equation containing a nonlinear dissipative effect is investigated the existence of the weak solution of the equation in lower order sobolev space hs with is established by using the techniques of the pseudoparabolic regularization and some a priori estimates derived from the equation itself

infinite propagation speed for the degasperis procesi equation abstract we prove that any nontrivial classical solution of the degasperis procesi equation will not have compact support if its initial data has this property

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

global existence of weak solutions for a shallow water equation a nonlinear shallow water equation which includes the famous camassa holm ch and degasperis procesi dp equations as special cases is investigated provided that initial value u email protected h s email protected email protected 32 u email protected l 1 r and email protected x 2 u 0 does not change sign it is shown that there exists a unique global weak solution to the equation

blow up solutions for the modified b family of equations abstract in this paper we firstly derive the blow up criterion and the lower bound of the maximal time of existence to the modified b family of equations the new blow up mechanism for the solutions with certain initial profiles and various parameters is then described in detail

the local well posedness and existence of weak solutions for a generalized camassa holm equation abstract a generalization of the camassa holm equation a model for shallow water waves is investigated using the pseudoparabolic regularization technique its local well posedness in sobolev space h s r with s 3 2 is established via a limiting procedure in addition a sufficient condition for the existence of weak solutions of the equation in lower order sobolev space h s with 1 s 3 2 is developed

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

a multivariate faa di bruno formula with applications a multivariate faa di bruno formula for computing arbitrary partial derivatives of a function composition is presented it is shown by way of a general identity how such derivatives can also be expressed in the form of an infinite series applications to stochastic processes and multivariate cumulants are then delineated

gevrey micro regularity for solutions to first order nonlinear pde abstract let x t r m r and u c 2 r m r we study the gevrey micro regularity of solutions u of the nonlinear equation u t f x t u u x where f x t 0 is a gevrey function of order s 1 and holomorphic in 0 we show that the gevrey wave front set of any c 2 solution u is contained in the characteristic set of the linearized operator l u t j 1 m f j x t u u x x j to achieve this we study the notion of gevrey approximate solutions a concept which we believe is of independent interest and could be applied to much more general situations

geometric finite difference schemes for the generalized hyperelastic rod wave equation geometric integrators are presented for a class of nonlinear dispersive equations which includes the camassa holm equation the bbm equation and the hyperelastic rod wave equation one group of schemes is designed to preserve a global property of the equations the conservation of energy while the other one preserves a more local feature of the equations the multi symplecticity

grade two fluids on non smooth domain driven by multiplicative noise existence uniqueness and regularity the author presented a lecture on the subject of this paper at the workshop on numerics for stochastic partial differential equations and their applications held at the johann radon institute for computational and applied mathematics ricam linz austria during the period 04 16 december 2016

exponential stability and inequalities of solutions of abstract functional differential equations abstract with the lyapunov second method we study the abstract functional differential equation d u d t f t u t we obtain inequalities of solutions and exponential stability with conditions like i w 1 u t x v t u t w 2 d t u t t h t l s w 1 u s x d s ii v 1 t u t t w 2 d t u t p t applications in ordinary and partial functional differential equations are given

the periodic b equation and euler equations on the circle in this note we show that the periodic b equation can only be realized as a euler equation on the lie group diff s1 of all smooth and orientation preserving diffeomorphisms on the circle if b 2 i e for the camassa holm equation in this case the inertia operator generating the metric on diff s1 is given by a 1 x2 in contrast the degasperis procesi equation for which b 3 is not a euler equation on diff s1 for any inertia operator our result generalizes a recent result of kolev some geometric investigations on the degasperis procesi shallow water equation wave motion 46 412 419 2009

the fbi transforms and their use in microlocal analysis abstract using a more general class of fbi transforms introduced by s berhanu and j hounie in 12 we completely characterize regularity and microregularity in denjoy carleman non quasi analytic classes which includes the gevrey classes and m christ version of the fbi transform defined in 22 as examples we also exhibit a result on microlocal regularity for solutions of first order partial differential equations in these classes that do not seem possible to prove using the classical fbi transform

integrable hierarchies related to the kuper ch spectral problem in this paper from a given kuper ch spectral problem we propose two kinds of super integrable hierarchies one is the kuper ch hierarchy the other is the generalized kuper harry dym hierarchy moreover we construct their zero curvature representations and super bi hamiltonian structures

on the curvature blow up phenomena for the fokas qiao xia li equation abstract we study the blow up phenomena of the strong solutions for the fokas qiao xia li fqxl equation with quadratic and cubic nonlinearities which include the celebrated camassa holm equation and the fokas olver rosenau qiao forq equation as its special case the fqxl equation models the unidirectional propagation of the shallow water waves over a flat bottom and describes the physics more accurately than its celebrated counterpart the gardner equation two kinds of conditions on the initial data including cases where the initial momentum density is not of one sign are formulated to guarantee finite time blow up for the strong solutions for this equation the first condition relies on the sign persistence property of the momentum density m the second condition is based on the full use of the conservation laws of the equation especially the conservation quantity h 2 u 0

quasianalytic denjoy carleman classes and o minimality we show that the expansion of the real field generated by the functions of a quasianalytic denjoy carleman class is model complete and o minimal provided that the class satisfies certain closure conditions some of these structures do not admit analytic cell decomposition and they show that there is no largest o minimal expansion of the real field