a method for constructing nontraveling wave solutions for 1 1 dimensional evolution equations a method is proposed to construct nontraveling wave solution for 1 1 dimensional evolution equations by extending the linear traveling wave transformation of the g g expansion method into a nonlinear transformation owing to a built in arbitrary function included in the solution abundant solutions can be excited taking the vakhnenko equation as an example a series of nontraveling wave solutions with variable separation is obtained some new solutions are excited and the known solitary wave solutions are special cases of the nontraveling wave solutions

the sine cosine method for obtaining solutions with compact and noncompact structures in this paper we establish compact and noncompact solutions for nonlinear dispersive equations a sine cosine method is used to demonstrate this work the different physical structures of the focusing branch and the defocusing branch are emphasized many models are approached to illustrate the usage of our main results

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

serre weights for rank two unitary groups we study the weight part of a generalisation of serre s conjecture for mod l galois representations associated to automorphic representations on rank two unitary groups for odd primes l we propose a conjectural set of serre weights agreeing with all conjectures in the literature and under a mild assumption on the image of the mod l galois representation we are able to show that any modular representation is modular of each conjectured weight we make no assumptions on the ramification or inertial degrees of l our main innovations are the use of outer forms of text gl 2 to avoid the parity difficulties with inner forms and the use of the lifting techniques and change of weight results introduced in barnet lamb et al j am math soc 24 2 411 469 2011 duke math j 161 8 1521 1580 2012 potential automorphy and change of weight preprint 2010

a note on the g g expansion method abstract we demonstrate that the g g expansion method which is often used in finding exact solutions of nonlinear differential equation is equivalent to the well known tanh method and application of these methods gives the same exact solutions of nonlinear differential equations

exact solutions of the bogoyavlenskii equation using the multiple g g expansion method in this work exact traveling wave solutions of the bogoyavlenskii equation are studied by the g g expansion method and its variants the solutions obtained are general solutions which are in the form of hyperbolic trigonometric and rational functions and a variety of special solutions like kink shaped antikink shaped bell type soliton solutions etc can easily be derived from the general results under certain domain these methods are more effective and simple than other methods and a number of solutions can be obtained at the same time in order to visualize the underlying dynamics of the obtained solutions two and three dimensional plots are drawn

soliton solutions and backlund transformation for the complex ginzburg landau equation abstract symbolically investigated in this paper is the complex ginzburg landau cgl equation with the hirota method both bright and dark soliton solutions for the cgl equation are obtained simultaneously new backlund transformation in the bilinear form is derived relevant properties and features are discussed solitons can be compressed amplified when the nonlinear linear dispersion effect is enhanced meanwhile central frequency of the soliton can be affected by the nonlinear and linear dispersion effects besides directions of the movement for the soliton central frequency can be adjusted results of this paper would be of certain value to the studies on the soliton compression and amplification

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

the fontaine mazur conjecture for gl _2 introduction 641 1 breuil mezard conjecture and the p adic local langlands 644 1 1 the breuil mezard conjecture 644 1 2 review of colmez s functor 647 1 3 hilbert samuel multiplicities 650 1 4 representations and pseudo representations 653 1 5 gl2 qp representations mod p 657 1 6 local patching and multiplicities 661 1 7 from pseudo representations to representations 669 2 modularity via the breuil mezard conjecture 675 2 1 quaternionic forms 675 2 2 global patching and multiplicities 677 2 3 the breuil mezard conjecture 686 addendum to ki 1 687 references 689

application of the g g expansion to travelling wave solutions of the broer kaup and the approximate long water wave equations by using the g g expansion proposed recently the travelling wave solutions involving parameters of the broer kaup equations and the approximate long water wave equations are found out the travelling wave solutions are expressed by three types of functions which are the hyperbolic functions the trigonometric functions and the rational functions when the parameters are taken as special values the solitary wave solutions are obtained c 2008 elsevier inc all rights reserved

integral crystalline cohomology over very ramified valuation rings in this paper we intend to show how fontaine s comparison theory fol fo2 between crystalline and p adic etale cohomology can be extented to very ramified base rings so far it had been developed either for qp coefficients or zp coefficients if the base is unramified over the witt vectors see fa2 for example it states that for any p adic discrete valuation ring v there exists a ring b v with the following property if x denotes a smooth and proper v scheme the p adic etale cohomology of the generic fiber x xv k is related to the crystalline cohomology of x v0 vo defined below by an isomorphism

exact solutions of nonlinear diffusion equations by variational iteration method in this paper he s variational iteration method is applied to obtain exact solutions of some nonlinear diffusion equations the variational iteration method is used to construct correction functionals using general lagrange multipliers identified optimally via the variational theory and the initial approximations can be freely chosen with unknown constants the solutions obtained are compared with those obtained by the adomian decomposition method and homotopy perturbation method showing excellent agreement but the variational iteration method is more effective he s variational iteration method can be introduced to overcome the difficulties arising in calculating adomian polynomials

searching for an agent who may or may not want to be found there is an extensive theory regarding optimal continuous path search for a mobile or immobile target the traditional theory assumes that the target is one of three types i an object with a known distribution of paths ii a mobile or immobile hider who wants to avoid or delay capture or iii a rendezvouser who wants to find the searcher this paper introduces a new type of search problem by assuming that aims of the target are not known to the searcher the target may be either a type iii cooperator with a known cooperation probabilityc or a type ii evader this formulation models search problems like that for a lost teenager who may be a runaway or a lost intelligence agent who may be a defector in any given search context it produces a continuum of search problems g c 0 c 1 linking a zero sum searchgame withc 0 to a rendezvous problem withc 1 these models thus provide a theoretical bridge between two previously distinct parts of search theory namely searchgames and rendezvous search

on the modularity of 2 adic potentially semi stable deformation rings using p adic local langlands correspondence for text gl 2 mathbb q 2 and an ordinary r mathbb t theorem we prove that the support of patched modules for quaternionic forms meet every irreducible component of the potentially semi stable deformation ring this gives a new proof of the breuil mezard conjecture for 2 dimensional representations of the absolute galois group of mathbb q 2 which is new in the case overline r a twist of an extension of the trivial character by itself as a consequence a local restriction in the proof of fontaine mazur conjecture in pask nas algebra number theory 10 6 1301 1358 2016 is removed

serre s modularity conjecture ii we provide proofs of theorems 4 1 and 5 1 of khare and wintenberger invent math doi 10 1007 s00222 009 0205 7 2009

multiplicites modulaires et representations de rm gl sb 2 mathbf z sb p et de rm gal overline mathbf q sb p mathbf q sb p en ell p appendice par guy henniart sur l unicite des types pour rm gl sb 2

deforming semistable galois representations abstract let v be a p adic representation of gal q q one of the ideas of wiles s proof of flt is that if v is the representation associated to a suitable autromorphic form a modular form in his case and if v is another p adic representation of gal q q closed enough to v then v is also associated to an automorphic form in this paper we discuss which kind of local condition at p one should require on v and v in order to be able to extend this part of wiles s methods

modularity lifting theorems for ordinary galois representations we generalize results of clozel harris and taylor by proving modularity lifting theorems for ordinary l adic galois representations of any dimension of an imaginary cm or totally real number field the main theorems are obtained by establishing an r mathrm red mathbb t theorem over a hida family a key part of the proof is to construct appropriate ordinary lifting rings at the primes dividing l and to determine their irreducible components

exact and explicit solutions to some nonlinear evolution equations by utilizing the g g expansion method in this paper we demonstrate the effectiveness of the so called g g expansion method by examining some nonlinear evolution equations with physical interest our work is motivated by the fact that the g g expansion method provides not only more general forms of solutions but also periodic and solitary waves if we set the parameters in the obtained wider set of solutions as special values then some previously known solutions can be recovered the method appears to be easier and faster by means of a symbolic computation system

observations on the tanh coth expansion method for finding solutions to nonlinear evolution equations abstract the tanh coth expansion method for finding solitary travelling wave solutions to nonlinear evolution equations has been used extensively in the literature it is a natural extension to the basic tanh function expansion method which was developed in the 1990s it usually delivers three types of solution namely a tanh function expansion a coth function expansion and a tanh coth expansion it is known that for every tanh function expansion solution there is a corresponding coth function expansion solution it is shown that there is a tanh coth expansion solution that is merely a disguised version of the coth solution in many papers such tanh coth solutions are erroneously claimed to be new however other tanh coth solutions may be delivered that are genuinely new in the sense that they would not be delivered via the basic tanh function method similar remarks apply to tan cot and tan cot expansion solutions

on the breuil mezard conjecture we give a new local proof of the breuil mezard conjecture for 2 dimensional representations of the absolute galois group of qp when p 5 and the representation has scalar endomorphisms

automorphic lifts of prescribed types we prove a variety of results on the existence of automorphic galois representations lifting a residual automorphic galois representation we prove a result on the structure of deformation rings of local galois representations and deduce from this and the method of khare and wintenberger a result on the existence of modular lifts of specified type for galois representations corresponding to hilbert modular forms of parallel weight 2 we discuss some conjectures on the weights of n dimensional mod p galois representations finally we use recent work of taylor to prove level raising and lowering results for n dimensional automorphic galois representations

potentially semi stable deformation rings let k qp be a finite extension and gk gal k k the galois group of an alge braic closure k let f be a finite field of characteristic p and vf a finite dimensional f vector space equipped with a continuous action of gkthe study of the defor mation theory of galois representations was initiated by mazur ma who showed that if vf has no non trivial endomorphisms then it admits a universal deformation ring ryw after the work of wiles wi and the conjectures of fontaine mazur fm it became clear that for arithmetic applications it was important to understand cer tain quotients of ry corresponding to deformations satisfying certain conditions for example wiles uses deformations which arise from finite flat group schemes and the corresponding quotient of ryw was constructed by ramakrishna ra suppose that l k is a finite extension and let a b be integers it seems to be a kind of folklore conjecture that there should be a quotient of ry whose points in finite extensions of w f l p correspond to deformations of vf which become semi stable over l and have hodge tate weights in the interval a b this is closely related to the final conjecture of fo 3 special cases of this when vf is 2 dimensional are conjectured in the papers of fontaine mazur fm p 191 breuil conrad diamond taylor bcdt conj 1 1 1 and breuil m zard bm 2 2 2 4 the purpose of this paper is to prove such a result

exact travelling wave solutions of some nonlinear equations by g g expansion method abstract here we find exact solutions of some nonlinear evolution equations within the framework of the g g expansion method exact solutions of five nonlinear equations of physical importance viz the coupled schrodinger kdv equation the coupled nonlinear reaction diffusion equation the foam drainage equation the phi four equation and the dodd bullough mikhailov equation are obtained these general solutions can be reduced in some standard results derived by some other methods two and three dimensional plots of some of the results are also presented

application of the g g expansion method to kawahara type equations using symbolic computation abstract in this paper kawahara type equations are selected to illustrate the effectiveness and simplicity of the g g expansion method with the aid of a symbolic computation system three types of more general traveling wave solutions including hyperbolic functions trigonometric functions and rational functions with free parameters are constructed solutions concerning solitary and periodic waves are also given by setting the two arbitrary parameters involved in the traveling waves as special values

on 2 adic deformations we compute the versal deformation ring of a split generic 2 dimensional representation chi 1 oplus chi 2 of the absolute galois group of mathbb q p as an application we show that the breuil mezard conjecture for both non split extensions of chi 1 by chi 2 and chi 2 by chi 1 implies the breuil mezard conjecture for chi 1 oplus chi 2 the result is new for p 2 the proof works for all primes

solitary solutions of some nonlinear evolution equations in this paper the traveling wave reduction combined with the homogeneous balance method hbm is used to find new exact solutions by using less studied nonlinear partial differential equations npdg of higher order the usual starting point is a special transformation converting the npdg in its two variables x and t into an nonlinear ordinary differential equation node in the single variable x using the hyperbolic tangent method new exact solutions for the three npdgs are studied the new feature of this paper is of course the fact that we are dealing with npdgs which cannot be found by studying the relevant literature therefore we believe it is time for publishing some special solutions of this interesting kind of equations clearly we do not use a new formalism but with the aid of the tanh method we are able to calculate at least exact traveling wave solutions solitary waves which can physically interpreted as an action of wave propagation simultaneous we point out the necessity of such sophisticated methods since a general theory of npdgs does not exist the existence of nontrivial noncomplex valued solutions of the ramani equation enables us to generate families of real valued solutions of a nonlinear sixth order system special differential transformations are leading to connections between traveling wave solutions of associated nonlinear evolution equations

towards a unified method for exact solutions of evolution equations an application to reaction diffusion equations with finite memory transport we present a brief report on the different methods for finding exact solutions of nonlinear evolution equations explicit exact traveling wave solutions are the most amenable besides implicit and parametric ones it is shown that most of methods that exist in the literature are equivalent to the generalized mapping method that unifies them by using this method a class of formal exact solutions for reaction diffusion equations with finite memory transport is obtained attention is focused to the finite memory transport fisher and nagumo equations

application of g g expansion method to regularized long wave rlw equation in this paper the g g expansion method with the aid of maple is used to obtain a generalized soliton solution for the generalized regularized long wave rlw equation each of the obtained solutions namely hyperbolic function solutions and trigonometric function solutions contain an explicit linear function of the variables in the considered equation it is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering problems

sur la densite des representations cristallines de text gal overline mathbb q _p mathbb q _p let mathfrak x d be the p adic analytic space classifying the semisimple continuous representations text gal overline mathbb q p mathbb q p rightarrow mathrm gl d overline mathbb q p we show that the crystalline representations are zarski dense in many irreducible components of mathfrak x d including the components made of residually irreducible representations this extends to any dimension d previous results of colmez and kisin for d 2 for this we construct an analogue of the infinite fern of gouvea mazur in this context based on a study of analytic families of trianguline varphi gamma modules over the robba ring we show in particular the existence of a universal family of framed regular trianguline varphi gamma modules as well as the density of the crystalline varphi gamma modules in this family these results may be viewed as a local analogue of the theory of p adic families of finite slope automorphic forms and they are new already in dimension 2 the technical heart of the paper is a collection of results about the fontaine herr cohomology of families of trianguline varphi gamma modules

on crystabelline deformation rings of mathrm gal overline mathbb q _p mathbb q _p with an appendix by jack shotton we prove that certain crystabelline deformation rings of two dimensional residual representations of mathrm gal overline mathbb q p mathbb q p are cohen macaulay as a consequence this allows to improve kisin s r 1 p mathbb t 1 p theorem to an r mathbb t theorem

weight elimination in serre type conjectures we prove the weight elimination direction of the serre weight conjectures as formulated by herzig for forms of u n which are compact at infinity and split at places dividing p in generic situations that is we show that all modular weights for a mod p galois representation are contained in the set predicted by herzig under some additional hypotheses we also show modularity of all the obvious weights

a 2 adic automorphy lifting theorem for unitary groups over cm fields we prove a minimal type automorphy lifting theorem for 2 adic galois representations of unitary type over imaginary cm fields we use this to improve an automorphy lifting theorem of kisin for mathrm gl 2

potential automorphy and change of weight we prove an automorphy lifting theorem for l adic representations where we impose a new condition at l which we call potentential diagonalizability this result allows for change of weight and seems to be substantially more exible than previous theorems along the same lines we derive several applications for instance we show that any irreducible totally odd essentially self dual regular weakly compatible system of l adic representations of the absolute galois group of a totally real eld is potentially automorphic and hence is pure and its l function has meromorphic continuation to the whole complex plane and satises the expected functional equation

exact traveling solutions of some nonlinear evolution equation by g g expansion method in this paper we establish exact solutions for some nonlinear evolution equations the g g expansion method was used to construct traveling wave solutions of ramani s equations in this method we take the advantages of the general solutions of the second order lode to solve effectively many nonlinear evolution equations the g g expansion method presents a wider applicability for handling nonlinear wave equations

period spaces for hodge structures in equal characteristic we develop the analog in equal positive characteristic of fontaine s theory for crystalline galois representations of a p adic eld in particular we describe the analog of fontaine s functor which assigns to a crystalline galois representation an isocrystal with a hodge ltration in equal characteristic the role of isocrystals and hodge ltrations is played by z isocrystals and hodge pink structures the latter were invented by pink our rst main result in this article is the analog of the colmez fontaine theorem that weakly admissible implies admissible next we construct period spaces for hodge pink structures on a xed z isocrystal these period spaces are analogs of the rapoport zink period spaces for fontaine s ltered isocrystals in mixed characteristic and likewise are rigid analytic spaces for our period spaces we prove the analog of a conjecture of rapoport and zink stating the existence of a universal local system on a berkovich open subspace of the period space as a consequence of weakly admissible implies admissible this berkovich open subspace contains every classical rigid analytic point of the period space as the principal tool to demonstrate these results we use the analog of kedlaya s slope filtration theorem which we also formulate and prove here

exact travelling solutions for the lax s seventh order kdv equation by sech method and rational exp function method in this paper we establish exact solutions for a nonlinear evolution equation the sech method and the exp function method are used to construct the solitary travelling wave solutions of lax s seventh order kdv equation these solutions may be important of significance for the explanation of some practical physical problem