change of rings and singularity categories abstract we investigate the behavior of singularity categories and stable categories of gorenstein projective modules along a morphism of rings the natural context to approach the problem is via change of rings that is the classical adjoint triple between the module categories in particular we identify conditions on the change of rings to induce functors between the two singularity categories or the two stable categories of gorenstein projective modules moreover we study this problem at the level of big singularity categories in the sense of krause 30 along the way we establish an explicit construction of a right adjoint functor between certain homotopy categories this is achieved by introducing the notion of 0 cocompact objects in triangulated categories and proving a dual version of bousfield s localization lemma we provide applications and examples illustrating our main results

spectra of modules abstract this manuscript solves the problem that the so called stable category mod of an artin algebra is in general not triangulated the method is to mimick topology and hence first form the spanier whitehead category stab mod and then construct a category spectra of spectra of modules which completes the compact part of stab mod under small coproducts spectra is then a triangulated substitute for mod the main results are that spectra is a compactly generated triangulated category which contains the compact part of stab mod as a full subcategory and even admits a precise description of its compact objects which only form a small set of isomorphism classes as an application it is proved that over an artin algebra the gorenstein projective modules form a pre covering class this was previously only known for rings satisfying strong homological conditions

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

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

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

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

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

painleve test and the first painleve hierarchy starting from the first painleve equation painleve type equations of higher order are obtained by using the singular point analysis

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

schlesinger transformations of painleve ii v the explicit form of the schlesinger transformations for the second third fourth and fifth painleve equations is given

new integrable equations of fourth order and higher degree related to cosgrove s equation we give a general formulation of the algorithm of fokas and ablowitz which then allows us to obtain transformations for nth order ordinary differential equations to equations of the same order but perhaps of higher degree previously this algorithm has been used to obtain transformations for the six second order equations defining new transcendental functions discovered by painleve and co workers either to other equations in the painleve classification or to equations of second order and second degree as an example of our approach we consider a new fourth order ordinary differential equation due to cosgrove which is believed to define a new transcendent we obtain transformations relating this equation to other fourth order ordinary differential equations of degrees 2 all of these transformations as well as the corresponding higher degree differential equations all of which have the painleve property are new

nonequilibrium phenomena outlines and bibliographies of a workshop the following is a set of outlines and bibliographies for lectures presented at a summer workshop on nonequilibrium phenomena held from june 22 to july 3 1981 at the institute for theoretical physics in santa barbara these outlines were distributed to the participants in lieu of formal proceedings and they are being presented for publication in the same form in the belief that the information they contain will be useful to a wider audience it should be clearly stated however that the compilation is an informal one which does not claim to be a complete survey of the subject

gorenstein homological dimensions and auslander categories abstract in this paper we study gorenstein projective and flat modules over a noetherian ring r for an r module m we show that gorenstein projective dimension of m is finite if and only if gorenstein flat dimension of m is finite provided the krull dimension of r is finite moreover in the case that r is local we prove that gorenstein projective dimension of an r module m is finite if and only if r r m belongs to the auslander category of r

semi dualizing complexes and their auslander categories let r be a commutative noetherian ring we study r modules and complexes of such with excellent duality properties while their common properties are strong enough to admit a rich theory we count among them such potentially diverse objects as dualizing complexes for r on one side and on the other the ring itself in several ways these two examples constitute the extremes and their well understood properties serve as guidelines for our study however also the employment in recent studies of ring homomorphisms of complexes lying between these extremes is incentive

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

local well posedness and blow up phenomena of the generalized short pulse equation in this paper we investigate the cauchy problem of the generalized short pulse equation in periodic domain we first establish the local well posedness of the equation in hs s s 2 by taking advantage of the kato method and present some useful conservation laws then we obtain the boundedness of the solution by virtue of the conservation laws finally we give a precise blow up criterion for 0 or 0 and get several blow up results provided the initial data satisfies some certain conditions

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

convection front propagation and wavenumber selection abstract the propagation of convection front in a horizontal fluid layer heated from below is studied by the numerical integration of both the newell whitehead segel amplitude equation and the full boussinesq equations it is shown that the propagation speed before its steady state value is reached can lie within a wide range and vary depending on the width of the front profile time dependent regimes play an important role resulting in different wavenumbers of the roll pattern formed behind the front the rate of roll relaxation or tendency toward the optimal preferred wavenumber strongly depends on the width of the region occupied by the roll pattern and on the front propagation speed at large speeds the freezing of the wavenumbers is possible the effect of the rayleigh number r is mainly indirect by the variation of the speed of the front so that at lower r the relaxation is faster the evolution of the whole pattern may be complicated because of the combined effect of two competing mechanisms the role relaxation and the counteraction of the mass of other rolls the available results of the studies of front propagation do not contradict the possibility of the existence of the preferred wavenumber

effects of boundaries on one dimensional reaction diffusion equations near threshold a simple set of two reaction diffusion equations is analyzed near the threshold for the appearance of spatially periodic solutions in one dimension exact amplitude equations are derived to second order in the deviation from the threshold wavevector selection is studied analytically for parameters varying slowly in space from supercritical to subcritical conditions the nonuniversality of the selection process is demonstrated explicitly for nonpotential cases the effect of boundary conditions on restricting the available band of wavevectors is studied for solutions which fall below their bulk value near the boundary type i the existence of another class of static solutions type ii remaining finite at the boundary even near threshold is demonstrated a linear stability analysis of the uniform state reveals that type i solutions are often unstable immediately above threshold then one has type ii states or oscillatory solutions some numerical results are presented which confirm the analytic calculations

second order fourth degree painleve type equations transformations that involve a fuchsian type equation are used to obtain one to one correspondence between the painleve i iv equations and certain second order fourth degree painleve type equations

quillen equivalences for stable categories abstract for an abelian category a we investigate when the stable categories gpro j a and gin j a are triangulated equivalent to this end we realize these stable categories as homotopy categories of certain non trivial model categories and give conditions on a that ensure the existence of a quillen equivalence between the model categories in question we also study when such a quillen equivalence transfers from a to the category of chain complexes in a

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

a strange recursion operator demystified we show that a new integrable two component system of kdv type studied by karasu kalkanli et al 2004 acta appl math 83 85 94 is bi hamiltonian and its recursion operator which has a highly unusual structure of nonlocal terms can be written as a ratio of two compatible hamiltonian operators found by us using this we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution and the evolution systems corresponding to these symmetries are bi hamiltonian as well we also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form with the integration operator d 1x appearing in each term at most once

evolution of reaction diffusion patterns in infinite and bounded domains abstract we introduce a semi analytical method to study the evolution of spatial structures in reaction diffusion systems it consists in writing an integral equation for the relevant densities from the propagator of the linear part of the evolution operator in order to test the method we perform an exhaustive study of a one dimensional reaction diffusion model associated to an electrical device the ballast resistor we consider the evolution of step and bubble shaped initial density profiles in free space as well as in a semi infinite domain with dirichlet and neumann boundary conditions the piecewise linear form of the reaction term which preserves the basic ingredients of more complex nonlinear models makes it possible to obtain exact wave front solutions in free space and stationary solutions in the bounded domain short and long time behaviour can also be analytically studied whereas the evolution at intermediate times is analyzed by numerical techniques we paid particular attention to the features introduced in the evolution by boundary conditions

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

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

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 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

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

second order second degree painleve equations related with painleve i vi equations and fuchsian type transformations one to one correspondence between the painleve i vi equations and certain second order second degree equations of painleve type is investigated the transformation between the painleve equations and second order second degree equations is the one involving the fuchsian type equation

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

an electrothermal instability the influence of albedo boundary conditions on the stationary states of an exactly solvable model abstract in order to study the effect of partially reflective albedo boundary conditions on pattern formation and stability in reaction diffusion systems we have analyzed an exactly soluble model of an electrothermal instability the ballast resistor the present results allow a continuous interpolation between the results corresponding to neumann and dirichlet boundary conditions and they allow qualitative changes in the patterns to be followed

experiments with pattern forming systems abstract six experimental investigations of pattern formation problems are reviewed they were chosen so as to illustrate in succession increasingly complex issues involving systems with spatial variation the first three deal with the influence of noise or stochastic forces on non equilibrium systems a second set of experiments addresses progressively more complicated deterministic nonlinear issues the first experiment which is discussed involves the response of a system below its first bifurcation point to external noise the particular example studied experimentally was electro convection in a nematic liquid crystal the noise induced fluctuations can be understood in terms of a linear theory the second experiment addresses the question of how stochastic effects manifest themselves in the pattern evolution which will take place when the control parameter of the system is increased above its threshold value this was investigated for the case of rayleigh benard convection rbc it involves a combination of linear and weakly non linear phenomena the third topic deals with stochastic effects on a stationary time periodic process in a pattern forming system namely rbc with periodic modulation of the rayleigh number the second set of experiments addresses questions for which stochastic effects play a negligible role this is so when the pattern amplitude is so large and bifurcations are so far away that the perturbations due to noise can be ignored the first topic in this category is an investigation of the band of stable states that exists above the first bifurcation in taylor vortex flow tvf the next issue is the selection of particular states out of this band by a spatial variation of the control parameter it turns out that this selection is determined by the nature of the control parameter variation and not by the non linear state as such thus this experiment failed to provide any evidence for the existence of an extremum principle which might control the pattern selection process however the results could be understood quantitatively in terms of theoretical models involving phase equations and thus provide a welcome test of this theoretical approach well above the bifurcation where amplitude equations are no longer applicable the third experiment in this category can be explained in terms of an amplitude equation but requires complex coefficients it involves the study of localized states or pulses of travelling convection rolls in a binary mixture these pulses are related to the solitons in dispersive systems but differ from them because of the dissipation in the hydrodynamic system the equivalent of these pulses could not exist in a thermodynamic system where the existence of a free energy would assure the survival of only one phase for a given set of intensive parameter values

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

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

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

third order differential equations with fixed critical points abstract the singular point analysis of third order ordinary differential equations which are algebraic in y and y is presented some new third order ordinary differential equations that pass the painleve test as well as the known ones are found

envelopes and covers by modules of finite injective and projective dimensions abstract in this paper we study the existence of l envelopes l envelopes d envelopes d covers and l covers where l and d denote the classes of modules of injective and projective dimension less than or equal to a natural number n respectively we prove that over any ring r special d preenvelopes and special d precovers always exist if the ring is noetherian the same holds for l envelopes and for d envelopes and d covers when the ring is perfect when inj dim r n then l covers exist and if r is such that a given class of homomorphisms is closed under well ordered direct limits then l envelopes exist

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

chazy s second degree painleve equations we examine two sets of second degree painlev equations derived by chazy in 1909 denoted by systems ii and iii the last member of each set is a second degree version of the painlev vi equation and there are no other second order second degree painlev equations in the polynomial class with this property we map the last member of system ii into the fokas yortsos equation and demonstrate how both schlesinger and okamoto transformations for painlev vi can be read off the chazy equation the 24 fundamental schlesinger transformations were known to garnier in 1943 while the 64 okamoto transformations date from 1987 in an appendix we gather together the solutions of the five members of system ii system iii is better known being equivalent to jimbo and miwa s equations for the logarithmic derivatives of the tau functions of the six painlev transcendents the last member known to painlev in 1906 was written in a manifestly symmetric form by jimbo and miwa suggesting many induced symmetries for painlev vi in particular schlesinger and okamoto transformations for painlev vi can be read off immediately

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

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

pattern competition and the decay of unstable patterns in quasi one dimensional systems abstract the dynamics of quasi one dimensional pattern forming systems with simple periodic initial conditions and additional noise or with almost periodic initial conditions is studied analytically and numerically using the lowest order amplitude equation in this way the decay of eckhaus unstable patterns and subsequent formation of stable ones as observed recently and the competition of patterns with different wavenumbers is studied

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

nonlinear differential equations of the second third and fourth order with exact solutions abstract we present the class of nonlinear ordinary differential equations of the second third and fourth order in a polynomial form with exact solutions this class has exact solutions in the form of solitary waves kinks and one periodic solutions

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

on traveling wave solutions of the θ equation of dispersive type abstract traveling wave solutions to a class of dispersive models u t u t x x u u x u u x x x 1 u x u x x are investigated in terms of the parameter including two integrable equations the camassa holm equation 1 3 and the degasperis procesi equation 1 4 as special models it was proved in h liu and z yin 2011 39 that when 1 2 1 smooth solutions persist for all time and when 0 1 2 strong solutions of the equation may blow up in finite time yielding rich traveling wave patterns this work therefore restricts to only the range 0 1 2 it is shown that when 0 only periodic travel wave is permissible and when 1 2 traveling waves may be solitary periodic or kink like waves for 0 1 2 traveling waves such as periodic solitary peakon peaked periodic cusped periodic or cusped soliton are all permissible

conservative weak solutions of the periodic cauchy problem for μhs equation we prove the existence and uniqueness of conservative weak solutions of the periodic cauchy problem for an integrable evolution equation from mathematical physics our method is to first prove the existence and uniqueness of solutions for all time in lagrangian coordinates and construct the maps to carry this result to the eulerian coordinates

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

second order second degree painleve equations related with painleve i ii iii equations the algorithmic method introduced by fokas and ablowitz to investigate the transformation properties of painleve equations is used to obtain a one to one correspondence between the painleve i ii and iii equations and certain second order second degree equations of painleve type

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

on the solvability of painleve ii and iv we introduce a rigorous methodology for studying the riemann hilbert problems associated with certain integrable nonlinear ordinary differential equations for concreteness we investigate the painleve ii and painleve iv equations we show that the cauchy problems for these equations admit in general global meromorphic in t solutions furthermore for special relations among the monodromy data and for t on stokes lines these solutions are bounded for finite t

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

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

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

on gorenstein projective injective and flat dimensions a functorial description with applications gorenstein homological dimensions are refinements of the classical homological dimensions and finiteness singles out modules with amenable properties reflecting those of modules over gorenstein rings as opposed to their classical counterparts these dimensions do not immediately come with practical and robust criteria for finiteness not even over commutative noetherian local rings in this paper we enlarge the class of rings known to admit good criteria for finiteness of gorenstein dimensions

schlesinger transformations for painleve vi equation a method to obtain the schlesinger transformations for painleve vi equation is given the procedure involves formulating a riemann hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same

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