global solutions and blow up phenomena to a shallow water equation abstract a nonlinear shallow water equation which includes the famous camassa holm ch and degasperis procesi dp equations as special cases is investigated the local well posedness of solutions for the nonlinear equation in the sobolev space h s r with s 3 2 is developed provided that 1 x 2 u 0 does not change sign u 0 h s s 3 2 and u 0 l 1 r the existence and uniqueness of the global solutions to the equation are shown to be true in u t x c 0 h s r c 1 0 h s 1 r conditions that lead to the development of singularities in finite time for the solutions are also acquired

finding an induced subdivision of a digraph we consider the following problem for oriented graphs and digraphs given an oriented graph digraph g does it contain an induced subdivision of a prescribed digraph d the complexity of this problem depends on d and on whether g must be an oriented graph or is allowed to contain 2 cycles we give a number of examples of polynomial instances as well as several np completeness proofs

finding a subdivision of a digraph we consider the following problem for oriented graphs and digraphs given a directed graph d does it contain a subdivision of a prescribed digraph f we give a number of examples of polynomial instances several np completeness proofs as well as a number of conjectures and open problems

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

long time behaviour for a class of low regularity solutions of the camassa holm equation in this paper we investigate the long time behaviour for a class of low regularity solutions of the camasssa holm equation given by the superposition of infinitely many interacting traveling waves with corners at their peaks

fredholm determinants and the camassa holm hierarchy the equation of camassa and holm 2 2 is an approximate description of long waves in shallow water

traveling wave solutions of the camassa holm equation all weak traveling wave solutions of the camassa holm equation are classified we show that in addition to smooth solutions there are a multitude of traveling waves with singularities peakons cuspons stumpons and composite waves

initial boundary value problems for nonlinear dispersive wave equations in this paper we study initial value boundary problems of two types of nonlinear dispersive wave equations on the half line and on a finite interval subject to homogeneous dirichlet boundary conditions we first prove local well posedness of the rod equation and of the b equation for general initial data furthermore we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span in the case of finite time singularities we are able to describe the precise blow up scenario of breaking waves our approach is based on sharp extension results for functions on the half line or on a finite interval and several symmetry preserving properties of the equations under discussion

disjoint directed and undirected paths and cycles in digraphs we show that the following problem is np complete given a digraph d and distinct vertices s t of d decide whether the underlying graph of d contains two internally disjoint s t paths p and q such that p is a directed s t path in d using this result we characterize those mixed linkage problems which are polynomially solvable assuming p np this complements the classical dichotomy by fortune hopcroft and wyllie classifying those directed linkage problems that are polynomially solvable we furthermore show that contrary to the case of directed linkages in digraphs the mixed problem remains np complete for acyclic digraphs

twisted holomorphic chains and vortex equations over non compact kahler manifolds abstract in this paper we study twisted holomorphic chains and related gauge equations over non compact kahler manifolds we use the heat flow method to solve the dirichlet boundary problem for the related gauge equations and prove a hitchin kobayashi type correspondence for twisted holomorphic chain over some non compact kahler manifolds

stability of peakons for the degasperis procesi equation the degasperis procesi equation can be derived as a member of a oneparameter family of asymptotic shallow water approximations to the euler equations with the same asymptotic accuracy as that of the camassaholm equation in this paper we study the orbital stability problem of the peaked solitons to the degasperis procesi equation on the line by constructing a liapunov function we prove that the shapes of these peakon solitons are stable under small perturbations

the limit of the hermitian yang mills flow on reflexive sheaves abstract in this paper we study the asymptotic behavior of the hermitian yang mills flow on a reflexive sheaf we prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the harder narasimhan seshadri filtration this answers a question by bando and siu

dirichlet problem for hermitian yang mills higgs equations over hermitian manifolds abstract in this paper we investigate the dirichlet problem for one type of vortex equations which generalize the well known hermitian yang mills equations over general hermitian manifolds

conservation laws of the camassa holm equation we use the bi hamiltonian structure of the camassa holm equation to show that its conservation laws h n m are homogeneous with respect to the scaling m lambdam moreover a direct argument is presented proving that h 1 h 2 are of local character finally simple representations of the conservation laws in terms of their variational derivatives are derived and used to obtain a constructive scheme for computation of the h n s

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

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

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

controllability and asymptotic stabilization of the camassa holm equation we investigate the problems of exact controllability and asymptotic stabilization of the camassa holm equation on the circle by means of a distributed control the results are global and in particular the control prevents the solution from blowing up

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

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

finding good 2 partitions of digraphs ii enumerable properties we continue the study initiated in 3 of the complexity of deciding whether a given digraph d has a vertex partition into two disjoint subdigraphs with given structural properties and given minimum cardinality let e be the following set of properties of digraphs e strongly connected connected minimum out degree at least 1 minimum in degree at least 1 minimum semi degree at least 1 minimum degree at least 1 having an out branching having an in branching in this paper we determine for all choices of p 1 p 2 from e and all pairs of fixed positive integers k 1 k 2 the complexity of deciding whether a digraph has a vertex partition into two digraphs d 1 d 2 such that d i has property p i and v d i k i i 1 2 we also classify the complexity of the same problems when restricted to strongly connected digraphs the complexity of the analogous problems when p 1 h and p 2 h e where h acyclic complete arc less oriented no 2 cycle semicomplete symmetric tournament were completely characterized in 3

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

global existence and blow up phenomena for the degasperis procesi equation this paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the degasperis procesi equation on the line global strong solutions to the equation are determined for a class of initial profiles on the other hand it is shown that the first blow up can occur only in the form of wave breaking a new wave breaking mechanism for solutions is described in detail and two results of blow up solutions with certain initial profiles are established

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

a complex frobenius theorem multiplier ideal sheaves and hermitian einstein metrics on stable bundles a complex frobenius theorem is proved for subsheaves of a holomorphic vector bundle satisfying a finite generation condition and a differential inclusion relation a notion of multiplier ideal sheaf for a sequence of hermitian metrics is defined the complex frobenius theorem is applied to the multiplier ideal sheaf of a sequence of metrics along donaldson s heat flow to give a construction of the destabilizing subsheaf appearing in the donaldson uhlenbeck yau theorem in the case of algebraic surfaces

global weak solutions and blow up structure for the degasperis procesi equation in this paper we study several qualitative properties of the degasperis procesi equation we first established the precise blow up rate and then determine the blow up set of blow up strong solutions to this equation for a large class of initial data we finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions

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

traveling wave solutions of the degasperis procesi equation we classify all weak traveling wave solutions of the degasperis procesi equation in addition to smooth and peaked solutions the equation is shown to admit more exotic traveling waves such as cuspons stumpons and composite waves

on the inverse spectral problem for the camassa holm equation abstract a key basis for seeking periodic solutions of the camassa holm equation is to understand the associated spectral problem y 1 4 y my the periodic spectrum can be recovered from the norming constants and the elements of the auxiliary spectrum the potential can then be reconstructed from the periodic spectrum a necessary and sufficient condition for exponential decrease of the widths 2n 2n 1for a sequence 0 m c1 r

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

global existence and blow up phenomena for the weakly dissipative camassa holm equation abstract this paper is concerned with global existence and blow up phenomena for the weakly dissipative camassa holm equation a new global existence result and a new blow up result for strong solutions to the equation with certain profiles are presented the obtained results improve considerable the previous results