local solution for the kadomtsev petviashvili equation in r2 abstract in this paper we obtain results about local existence uniqueness regularity and continuous dependence on the initial data for the solution of the kadomtsev petviashvili equation with positive dispersion in r 2 and initial data in sobolev spaces of order s 3 our method uses an approximated problem where the theory of quasi linear evolution equations is applicable

non existence of l 2 compact solutions of the kadomtsev petviashvili ii equation we prove that there is no nontrivial solution of the kadomtsev petviashvili ii equation kp ii equation u t u xxx uu x x u yy 0 quad x y in bf r 2 which is l 2 compact i e uniformly localized in l 2 norm and travel to the right in the x variable this result extends the previous work of de bouard and saut 3 stating that there is no traveling wave solution for the kp ii equation the proof uses a monotonicity property of the l 2 mass for solutions of the kp ii equation similar to the one for the kdv equation 12 14 and two virial type relations the result still holds for some natural generalizations of the kp ii equation general nonlinearity higher dispersion and does not rely on the integrability of the equation

on equations of kp type we discuss the local cauchy problem for the generalised kadomtsev petviashvili equation namely in the periodic and nonperiodic settings

global well posedness for the kp i equation on the background of a non localized solution we prove that the cauchy problem for the kp i equation is globally well posed for initial data which are localized perturbations of arbitrary size of a non localized i e not decaying in all directions traveling wave solution e g the kdv line solitary wave or the zaitsev solitary waves which are localized in x and y periodic or conversely

global well posedness in the energy space for a modified kp ii equation via the miura transform we prove global well posedness of the initial value problem for a modified kadomtsev petviashvili ii mkp ii equation in the energy space the proof proceeds in three main steps and involves several different techniques in the first step we make use of several linear estimates to solve a fourth order parabolic regularization of the mkp ii equation by a fixed point argument for regular initial data one estimate is similar to the sharp kato smoothing effect proved for the kdv equation by kenig ponce and vega 1991 then compactness arguments the energy method performed through the miura transform give the existence of a local solution of the mkp ii equation for regular data finally we approximate any data in the energy space by a sequence of smooth initial data using bourgain s result concerning the global well posedness of the kp ii equation in l 2 and the miura transformation we obtain convergence of the sequence of smooth solutions to a solution of mkp ii in the energy space

conservation laws of the 2 1 dimensional kp equation and burgers equation with variable coefficients and cross terms in this paper conservation laws for the 2 1 dimensional kp equation and burgers equation with variable coefficients and cross terms are studied due to the existence of cross terms the new conservation theorem given by ibragimov cannot be applied to the two equations directly we propose two rules of modifications which ensure the theorem can be applied to nonlinear evolution equations with cross terms formulas of conservation laws for the kp equation and burgers equation are given using these formulas non trivial and time dependent conservation laws for these equations are derived according to their different lie symmetries

asymptotics for large time of global solutions to the generalized kadomtsev petviashvili equation we study the large time asymptotic behavior of solutions to the generalized kadomtsev petviashvili kp equations where 1 or 1 when 2 and 1 kp is known as the kpi equation while 2 1 corresponds to the kpii equation the kp equation models the propagation along the x axis of nonlinear dispersive long waves on the surface of a fluid when the variation along the y axis proceeds slowly 10 the case 3 1 has been found in the modeling of sound waves in antiferromagnetics 15 we prove that if 3 is an integer and the initial data are sufficiently small then the solution u of kp satisfies the following estimates for all t r where 1 if 3 and 0 if 4 we also find the large time asymptotics for the solution

global well posedness for the fifth order kadomtsev petviashvili ii equation in three dimensional space abstract the cauchy problem for the fifth order kadomtsev petviashvili ii equation x t u x 5 u x u 2 y z u 0 is considered a small data global well posedness and scattering result in the scale and galilean invariant space h 1 2 0 0 r 3 is derived

bilinear space time estimates for linearised kp type equations on the three dimensional torus with applications abstract a bilinear estimate in terms of bourgain spaces associated with a linearised kadomtsev petviashvili type equation on the three dimensional torus is shown as a consequence time localized linear and bilinear space time estimates for this equation are obtained applications to the local and global well posedness of dispersion generalised kp ii equations are discussed especially it is proved that the periodic boundary value problem for the original kp ii equation is locally well posed for data in the anisotropic sobolev spaces h x s h y e t 3 if s 1 2 and e 0

the cauchy problem for the rotation modified kadomtsev petviashvili type equation abstract this paper is devoted to studying the cauchy problem for the rotation modified kadomtsev petviashvili rmkp type equation x u t x 3 u x u 2 y 2 u u 0 in the anisotropic sobolev spaces h s 1 s 2 r 2 when 0 and 0 0 we show that the cauchy problem is locally well posed in h s 1 s 2 r 2 with s 1 1 2 and s 2 0 the main difficulty in establishing bilinear estimates related to nonlinear term of rmkp type equation is that the resonant function 3 1 2 1 2 1 2 2 2 1 2 1 2 1 1 2 2 2 may tend to zero since 0 0 and 0 when 0 and 0 and 0 we also prove that the cauchy problem for rmkp equation is ill posed in h s 1 0 r 2 with s 1 1 2 in the sense that the flow map associated to the rotation modified kadomtsev petviashvili is not c 3

well posedness for the kadomtsev petviashvili ii equation and generalisations we show the local in time well posedness of the cauchy problem for the kadomtsev petviashvili ii equation for initial data in the non isotropic sobolev space h s 1 s 2 r 2 with s 1 j and s 2 0 on the h s 1 0 r 2 scale this result includes the full subcritical range without any additional low frequency assumption on the initial data more generally we prove the local in time well posedness of the cauchy problem for the following generalisation of the kp ii equation u t d x u x u 2 x x u yy 0 u 0 u 0 for 4 3 max 1 3 4 1 4 3 8 s 2 0 and u 0 h s 1 s 2 r 2 we deduce global well posedness for s 1 0 s 2 0 and real valued initial data

stability of the line soliton of the kp ii equation under periodic transverse perturbations we prove the nonlinear stability of the kdv solitary waves considered as solutions of the kp ii equation with respect to periodic transverse perturbations our proof uses a miura transform which sends the solutions of an mkp ii equation to solutions of the kp ii equation

nonextensive foundation of levy distributions a deep connection between the ubiquity of l evy distributions in nature and the nonextensive thermal statistics introduced a decade ago has been established recently tsallis et al phys rev lett 75 3589 1995 by using unnormalized q expectation values it has just been argued on physical grounds that normalized q expectation values should be used instead we revisit within this more appropriate scheme the l evy problem and verify that the relevant analytic results become sensibly simplified whereas the basic physics remains unchanged

long time bounds for the periodic benjamin ono bbm equation abstract we consider the periodic benjamin ono equation regularized in the same way as the benjamin bona mahony equation is found from the korteweg de vries one t b benjamin j l bona j j mahony model equations for long waves in nonlinear dispersive systems philos trans roy soc london ser a 272 1220 1972 47 78 namely the equation u t u x u u x h u x t 0 where h is the hilbert transform the quotient between the characteristic wave amplitude and the depth of the waves and the quotient between this depth and the wavelength we show that the solution starting from an initial datum of size e remains smaller than e for a time scale of order e 1 2 whereas the local well posedness gives only a time of order e 1

on the cauchy problem for a k p boussinesq type system abstract in this work we prove that the cauchy problem t u u x v u x u u t v v x uv u x 2 u x 1 y 2 u v u x y 0 x y v x y 0 x y is locally well posed for all and 0 in the anisotropic sobolev space ys s 2 for 0 we establish global well posedness for small data we also show the existence of solitary wave solutions for this system and prove regularity and analyticity of this kind of solutions

on the propagation of regularities in solutions of the fifth order kadomtsev petviashvili ii equation abstract in this paper we study special properties of solutions of the initial value problem ivp associated to the fifth order kadomtsev petviashvili ii equation mainly for initial data h 1 r 2 whose restriction belongs to h m x 0 r for some m z m 2 and x 0 r we prove that the restriction of the corresponding solution u t belongs to h m b r for any b r and any t 0 in other words the extra regularity on the data propagates in the solutions in the direction of the dispersion

on the extension of householder s method for weighted moore penrose inverse this note discusses how to extend and use the well known method of householder 1964 6 for finding weighted moore penrose inverse the discussions cover both theoretical and numerical aspects an approach for accelerating the initial phase of convergence will be contributed some tests will also be employed to check the validity and superiority of the results over the schulz method

q gaussian processes non commutative and classical aspects we examine for 1 q 1 q gaussian processes i e families of operators non commutative random variables where the a t fulfill the q commutation relations for some covariance function equipped with the vacuum expectation state we show that there is a q analogue of the gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q gaussian processes into corresponding and much simpler questions in the underlying hilbert space in particular we use this idea to show that a large class of q gaussian processes possesses a non commutative kind of markov property which ensures that there exist classical versions of these non commutative processes this answers an old question of frisch and bourret fb

a local well posed result for the fifth order kp ii initial value problem abstract the well posedness of the real valued fifth order kp ii equation in sobolev spaces h s 1 s 2 r 2 has been studied by several authors it was known to be locally well posed in h s 0 r 2 for s 5 4 see hadac 2008 3 isaza et al 2006 5 we are more interested in the case s 5 4 we obtain the local well posedness for s 1 5 4 s 2 0 which improves previous results by saut and tzvetkov 2000 15 and by isaza lopez and mejia 2006 5 moreover our main contribution is that we set up the global in time strichartz estimates for the fifth order kp equation no matter kp i or kp ii equation on dyadic pieces and especially on the low frequency part we obtain a b s l 4 estimate in which we obtain 3 8 regularity which comes from the effect of x 1 y 2 term

local regularity and decay estimates of solitary waves for the rotation modified kadomtsev petviashvili equation this paper is mainly concerned with the local low regularity of solutions and decay estimates of solitary waves to the rotation modified kadomtsevpetviashvili rmkp equation it is shown that with negative dispersion the rmkp equation is locally well posed for data in hs1 s2 r2 for s1 13 and s2 0 and hence globally well posed in the space l2 moreover an improved result on the decay property of the solitary waves is established which shows that all solitary waves of the rmkp equation decay exponentially at infinity

global well posedness for the kp i equation we prove the global well posedness of the kp i equation in bbb r 2 for data of arbitrary size in a suitable sobolev class we use a compactness method to get local solutions we extend the solutions thanks to some conservation laws an almost conserved quantity and strichartz estimates

low regularity for the fifth order kadomtsev petviashvili i type equation abstract the cauchy problem of the fifth order kadomtsev petviashvili i equation fifth kp i 0 1 t u x 5 u x 3 u x 1 y y u x u 2 0 x y t r 3 is considered it follows that the fifth order kadomtsev petviashvili i equation 0 1 is locally well posed in h s 0 with s 3 4 where the norm h s r is defined by f h x y s r s r f l 2

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

a q deformation of the gauss distribution the q deformed commutation relation aa qa a 1 for the harmonic oscillator is considered with q 1 1 an explicit representation generalizing the bargmann representation of analytic functions on the complex plane is constructed in this representation the distribution of a a in the vacuum state is explicitly calculated this distribution is to be regarded as the natural q deformation of the gaussian

on the asymptotic behavior of solutions to the generalized kadomtsev petviashvili burgers equations we show global existence and sharp results on the asymptotic decay of solutions to the generalized kp equations with a power nonlinearityupuxand with a burgers type dissipation in the main direction of propagation appended furthermore we prove that forp n the decay rate of thel2 norm of the solution of these equations is that exhibited by the solution of these same equations linearized around the zero solution and that forp 2 this also holds for thel norm this limitation seems due to the presence of a dispersive effect in the transverse direction

non extensive statistical mechanics and generalized fokker planck equation a non linear generalized fokker planck gfp equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by tsallis in his generalization of statistical mechanics in the case of a constant or linearly varying drift the time dependent solutions of the gfp equation are seen to obey a suitable form of the celebrated h theorem the temporal changes of tsallis entropies are seen to be given in terms of the fisher s information measure particular time dependent solutions of the gfp equation of the maximum tsallis entropy form are obtained

the cauchy problem and stability of solitary wave solutions for rlw kp type equations abstract the kadomtsev petviashvilli kp equation ut ux uux uxxx x euyy 0 arises in various contexts where nonlinear dispersive waves propagate principally along the x axis but with weak dispersive effects being felt in the direction parallel to the y axis perpendicular to the main direction of propagation we propose and analyze here a class of evolution equations of the form ut ux upux lut x euyy 0 which provides an alternative to eq in the same way the regularized long wave equation is related to the classical korteweg de vries kdv equation the operator l is a pseudo differential operator in the x variable p 1 is an integer and e 1 after discussing the underlying motivation for the class a local well posedness theory for the initial value problem is developed with assumptions on l and p that include conditions appertaining to models of interesting physical phenomenon the solutions defined locally in time t are shown to be smoothly extendable to the entire time axis in the particularly interesting case where l x2 and e 1 possesses travelling wave solutions u x y t c x ct y provided c 1 and 0 0 4 3 and c 1 and for 4 3 if c 4p 4 p the paper concludes with commentary on extensions of the present theory to more than two space dimensions

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

on the gaussian q distribution we present a study of the gaussian q measure introduced by diaz and teruel from a probabilistic and from a combinatorial viewpoint a main motivation for the introduction of the gaussian q measure is that its moments are exactly the q analogues of the double factorial numbers we show that the gaussian q measure interpolates between the uniform measure on the interval 1 1 and the gaussian measure on the real line 2009 elsevier inc all rights reserved

on the low regularity of the fifth order kadomtsev petviashvili i equation we consider the fifth order kadomtsev petviashvili i kp i equation as tu 3 x u 5 x u 1 x 2 y u uux 0 while r we introduce an interpolated energy space es to consider the well posedeness of the initial value problem ivp of the fifth order kp i equation we obtain the local well posedness of ivp of the fifth order kp i equation in es for 0 s 1 to obtain the local well posedness we present a bilinear estimate in the bourgain space in the framework of the interpolated energy space it crucially depends on the dyadic decomposed strichartz estimate the fifth order dispersive smoothing effect and maximal estimate

comments on the paper conservation laws of the 2 1 dimensional kp equation and burgers equation with variable coefficients and cross terms by li hua zhang the recent paper mentioned in the title contains a confusing statement on computing conservation laws corresponding to symmetries of nonlinearly self adjoint differential equations the present brief article contains clarifying comments

well posedness for the fifth order kp ii initial data problem in hs 0 r t abstract the well posed properties for the fifth order kp ii initial value problem for x r y t are considered it is proved to be locally well posed in h s 0 r t for s 3 4 with small initial data and s 3 4 with general initial data by the l 2 conservation law of kp equation the l 2 global well posedness is also obtained the crucial ingredient of the argument is the l 2 estimates of a bilinear operator which was introduced in recent works 14 and 13 this operator is galilean invariant in the content of t 2 and r 2 but not in the content r t

a uniqueness theorem for the two dimensional generalized korteweg de vries equation abstract using elementary techniques we prove that the solution which is periodic in both x and y to the two dimensional generalized kdv equation is unique a theorem of continuous dependence of the solutions on the initial data is established some estimates of the solutions are also included

large time asymptotics for the kadomtsev petviashvili equation we study the large time asymptotic behavior of solutions to the kadomtsev petviashvili equations left begin array ll u t u xxx sigma partial x 1 u yy partial x u 2 quad quad x y in bf r 2 t in bf r u 0 x y u 0 x y quad quad qquad qquad x y in bf r 2 end array right where 1 and partial x 1 int infty x dx prime we prove that the large time asymptotics of the derivative ux of the solution has a quasilinear character

classical versions of q gaussian processes conditional moments and bell s inequality we show that classical processes corresponding to operators which satisfy a q commutative relation have linear regressions and quadratic conditional variances from this we deduce that bell s inequality for their covariances can be extended from q 1 to the entire range 1 q 1

on periodic kp i type equations we analyze bilinear estimates related to periodic kp i type equations we prove that the cauchy problem for the fifth order kp i equation posed on is globally well posed in the energy space despite the loss of a smoothing relation which is essential to recuperate the derivative loss used in the context of the kdv and kp ii type equations we are able to gain a moothing thanks to some specific properties of the set of fourier modes where the smoothing relation fails counterexamples in the context of kp i type equations posed on are given these examples indicate that contrary to the kp ii case it would be difficult to obtain the local well posedness for the periodic kp i type equations by an iteration scheme using bourgain spaces

the cauchy problem for higher order kp equations abstract we study the local well posedness of higher order kp equations our well posedness results make an essential use of a global smoothing effect for the linearized equation established in ben artzi and saut preprint 1997 injected into the framework of fourier transform restriction spaces introduced by bourgain our ill posedness results rely on the existence of solitary wave solutions and on scaling arguments the method was first applied in the context of the kdv and schrodinger equations cf birnir et al ann inst h poincare anal non lineaire 13 1996 529 535 birnir et al j london math soc 53 1996 551 559

nonlinear self adjointness and conservation laws the general concept of nonlinear self adjointness of differential equations is introduced it includes the linear self adjointness as a particular case moreover it embraces the strict self adjointness definition 1 and quasi self adjointness introduced earlier by the author it is shown that the equations possessing nonlinear self adjointness can be written equivalently in a strictly self adjoint form by using appropriate multipliers all linear equations possess the property of nonlinear self adjointness and hence can be rewritten in a nonlinear strictly self adjoint form for example the heat equation ut u 0 becomes strictly self adjoint after multiplying by u 1 conservation laws associated with symmetries are given in an explicit form for all nonlinearly self adjoint partial differential equations and systems