average implicit linear difference scheme for generalized rosenau burgers equation abstract in this paper a linear three level average implicit finite difference scheme for the numerical solution of the initial boundary value problem of generalized rosenau burgers equation is presented existence and uniqueness of numerical solutions are discussed it is proved that the finite difference scheme is convergent in the order of o 2 h 2 and stable numerical simulations show that the method is efficient

chow groups of projective varieties of very small degree see 12 5 and the references given there these facts together with various conjectures on the cohomology and chow groups of algebraic varieties suggest that the chow groups of x might satisfy chl x q chl pk q q for l 1 compare with remark 5 6 and corollary 5 7 this is explicitly formulated by v srinivas and k paranjape in 16 conjecture 1 8 the chain of reasoning goes roughly as follows suppose x is smooth one expects a good filtration 0 f j 1 f j f 0 ch x x q whose graded pieces f f l 1 are controlled by h2j l x x see 10 according to grothendieck s generalized conjecture 8 the groups h x should be generated by the image under the gysin morphism of the homology of a codimension subset together with the classes coming from p applying this to the diagonal in x x should then force the triviality of the chow groups in the desired range for zero cycles the conjecture follows from roitman s theorem see 17 and 18

a better asymptotic profile of rosenau burgers equation this paper studies the large time behavior of the global solutions to the cauchy problem for the rosenau burgers r b equation u t u x x x x email protected x x u p 1 p 1 x 0 by the variable scaling method we discover that the solution of the nonlinear parabolic equation u email protected x x u p 1 p 1 x 0 is a better asymptotic profile of the r b equation the convergence rates of the r b equation to the asymptotic profile have been developed by the fourier transform method with energy estimates this result is better than the previous work 1 2 with zero as the asymptotic behavior furthermore the numerical simulations on several test examples are discussed and the numerical results confirm our theoretical results

a three level average implicit finite difference scheme to solve equation obtained by coupling the rosenau kdv equation and the rosenau rlw equation in the present work a mathematical model to obtain the solution of the nonlinear wave by coupling the rosenau kdv equation and the rosenau rlw equation is proposed the solution properties are also derived a numerical tool is applied to the model by using a three level average implicit finite difference technique the fundamental conservative properties of the equation are preserved by the presented numerical scheme and the existence and uniqueness of the numerical solution are proved moreover the convergence and stability of the numerical solution are also shown the new method give second order accurate in time and space thus the presented results can be constructed to demonstrate the viability of the new model

global positioning of robot manipulators via pd control plus a class of nonlinear integral actions deals with the position control of robot manipulators proposed is a simple class of robot regulators consisting of a linear proportional derivative pd feedback plus an integral action of a nonlinear function of position errors by using lyapunov s direct method and lasalle s invariance principle the authors characterize a class of such nonlinear functions and they provide explicit conditions on the regulator gains to ensure global asymptotic stability these regulators offer an attractive alternative to global regulation compared with the well known partially model based pd control with gravity compensation and pd control with desired gravity compensation

a linearly implicit conservative difference scheme for the generalized rosenau kawahara rlw equation this paper concerns the numerical study for the generalized rosenau kawahara rlw equation obtained by coupling the generalized rosenau rlw equation and the generalized rosenau kawahara equation we first derive the energy conservation law of the equation and then develop a three level linearly implicit difference scheme for solving the equation we prove that the proposed scheme is energy conserved unconditionally stable and second order accurate both in time and space variables finally numerical experiments are carried out to confirm the energy conservation the convergence rates of the scheme and effectiveness for long time simulation

numerical scheme for a model of shallow water waves in 2 1 dimensions abstract a nonlinear difference scheme is considered for the two dimensional rosenau burgers equation some priori estimates existence and uniqueness of the difference solution have been shown a second order convergence in the uniform norm and stability are proved also a convergent iterative algorithm is presented all results are obtained without any restrictions on the meshsizes at last numerical experiments are carried out to support the theoretical claims

unirational threefolds with no universal codimension 2 cycle we prove that the general quartic double solid with k le 7 nodes does not admit a chow theoretic decomposition of the diagonal or equivalently has a nontrivial universal mathrm ch 0 group and the same holds if we replace in this statement chow theoretic by cohomological in particular it is not stably rational we also deduce that the general quartic double solid with seven nodes does not admit a universal codimension 2 cycle parameterized by its intermediate jacobian and even does not admit a parametrization with rationally connected fibers of its jacobian by a family of 1 cycles this finally implies that its third unramified cohomology group is not universally trivial

examples of smash nilpotent cycles on rationally connected varieties abstract voevodsky has conjectured that numerical and smash equivalence coincide on a smooth projective variety we prove this conjecture holds for uniruled 3 folds 4 folds whose mrcc quotient has dimension 2 and for smooth complete intersections with very small degree

mixed finite element analysis for dissipative srlw equations with damping term abstract in this paper a mixed finite element mfe formulation is proposed for the initial boundary value problem of dissipative symmetric regularized long wave srlw equations with damping existence and uniqueness of its generalized solution and of the fully discrete mixed finite element solution are proved error estimates based on energy methods are given numerical experiments verify the theoretical analysis

rost nilpotence and higher unramified cohomology abstract we develop an approach to proving the rost nilpotence principle involving higher unramified cohomology along with a decomposition of the diagonal technique we use this to prove the principle for certain threefolds over a perfect field

a three level linear implicit conservative scheme for the rosenau kdv rlw equation abstract a three level linear conservative implicit finite difference scheme for solving the rosenau kdv rlw equation is proposed the existence uniqueness and conservation for energy of the difference solution are proved by the discrete energy method the new scheme is 2nd order convergent and unconditionally stable several numerical experiments verify that the theoretical results are accurate and the scheme is efficient and reliable

a generalised pid type control scheme with simple tuning for the global regulation of robot manipulators with constrained inputs in this paper a globally stabilising pid type control scheme with a generalised saturating structure for robot manipulators under input constraints is proposed it gives rise to various families of bounded pid type controllers whose implementation is released from the exact knowledge of the system parameters and model structure compared to previous approaches of the kind the proposed scheme is not only characterised by its generalised structure but also by its very simple tuning criterion the simplest hitherto obtained in the considered analytical framework experimental results on a 3 degree of freedom direct drive manipulator corroborate the efficiency of the proposed approach

a hyper kahler compactification of the intermediate jacobian fibration associated with a cubic 4 fold let x be a general cubic fourfold it was observed by donagi and markman that the relative intermediate jacobian fibration j u with u p5 x associated to the smooth hyperplane sections of x carries a natural holomorphic symplectic form making the fibration lagrangian in this paper we obtain a smooth projective compactification j of j with the property that the holomorphic symplectic form on j extends to a holomorphic symplectic form on j in particular j is a 10 dimensional compact hyper kahler manifold which we show to be deformation equivalent to the exceptional example of o grady this proves a conjecture by markushevich

the hodge conjecture for a certain class of singular varieties in j there is stated a version of the hodge conjecture for complex singular varieties reduced separated algebraic schemes over c generalizing the popular version of the conjecture a statement of the general hodge conjecture for the singular case extending the amended version by grothendieck is given in l2 using the techniques in j the statement is this let z be a reduced separated algebraic scheme over c the lowest weight part w i hi z q of the borelmoore homology hi z q defines a hodge structure of weight i recall the filtration by niveau given by

linearized difference schemes for a bbm equation with a fractional nonlocal viscous term this paper is concerned with the efficient finite difference schemes for a benjamin bona mahony equation with a fractional nonlocal viscous term by using the weighted shift grunwald letnikov and the fractional centered difference formulae to approximate the nonlocal fractional operators we design a class of linearized finite difference schemes for the presented nonlocal model the existence stability and convergence of the proposed numerical schemes are rigorously derived with the help of functional analysis theoretical analysis shows that the proposed numerical schemes are stable with second order accuracy numerical examples are presented to verify our theoretical analysis and to demonstrate the efficiency of the proposed numerical schemes

decent intersection and tor rigidity for modules over local hypersurfaces first published in transactions of the american mathematical society in 2013 published by the american mathematical society also available electronically from http www ams org journals tran 2013 365 06 s0002 9947 2012 05574 7 home html

the stability of feedback control with nonlinear saturating actuator time domain approach sufficient conditions for stability of closed loop systems with a saturating actuator are derived from the viewpoint of time domain the bellman gronwall inequality is used to investigate the stability of these systems three kinds of controllers are considered to treat this constrained control problem finally an example is given to illustrate the results

compact structure preserving approach to solitary wave in shallow water modeled by the rosenau rlw equation abstract a mass preserving scheme a nonlinear algorithm based on modification of a finite difference method to the rosenau rlw equation is proposed subject to homogeneous boundary conditions the key feature of the method for improving the accuracy of approximate solutions is to develop a compact higher order scheme together with an iterative algorithm for solving the nonlinear implicit scheme the derivatives for space discretization are approximated by using the algorithm dealing with a five point stencil in addition a three level average difference technique is used to perform time discretization the conservation of mass and both the existence and uniqueness of the numerical solution are proved the stability and convergence of the numerical solution with order o 4 2 h 2 h 4 are also confirmed for efficiency analysis numerical results show that the computational efficiency of the compact scheme is much higher than that of non compact schemes moreover long time behavior is also used to validate the capability of the present method

a new conservative finite difference scheme for the rosenau equation a conservative difference scheme is given for the kdv like rosenau equation the unique solvability of numerical solutions is shown a priori bound convergence and stability of the difference scheme are proved

convergence to diffusion waves of solutions for viscous conservation laws we study the large time behavior of solutions of viscous conservation laws it is shown that solutions tend to diffusion waves which are constructed based on the heat equation and burgers equation the convergence is in thelp 1 p sense and is obtained as a consequence of thel2 decay of the difference between the solution and its asymptotic state of diffusion waves

2 cycles on higher fano hypersurfaces let x d be a smooth hypersurface of degree d in mathbb p mathbb c n suppose that the fano variety mathrm f x d of lines of x d is smooth we prove that the griffiths group mathrm griff 1 mathrm f mathrm x d of mathrm f x d is trivial if the hypersurface mathrm x d is of 2 fano type as a result we give a positive answer to a question of professor voisin about the first griffiths groups of fano varieties in some cases base on this result we prove that mathrm ch 2 mathrm x d mathbb z for a smooth 3 fano hypersurface mathrm x d subseteq mathbb p n mathbb c with smooth fano variety of lines

a finite difference scheme for generalized regularized long wave equation in this paper a finite difference method for a cauchy problem of generalized regularized long wave grlw equation was considered an energy conservative finite difference scheme was proposed convergence and stability of the difference solution were proved numerical results demonstrate that the method is efficient and reliable

a conservative weighted finite difference scheme for regularized long wave equation in this paper a new finite difference method for initial boundary value problem of regularized long wave rlw equation is considered two energy conservative finite difference schemes are proposed and the prior estimate of the solution is obtained existence and uniqueness of numerical solution are derived it is proved that the finite difference schemes are convergent and stable numerical experiment results show that the theory is accurate and the schemes are efficient and reliable

lawson homology for varieties with small chow groups and the induced filtration on the griffiths groups these groups form an intriguing set of invariants despite the vast literature on this subject the groups have been calculated only in very few cases such as for the projective spaces one of the goals of this note is to understand these invariants in case the variety has few cycles in the sense that rational and homological equivalence coincide for all m cycles up to a certain rank m s actually the results only deal with that part of lawson homology that is not visible in ordinary homology in that it is in the kernel of the cycle class maps cm k 2m lmhk 2m x hk 2m x

development of multivariable pid controller gains in presence of measurement noise abstractthis paper examines the ability of a multivariable pid controller rejecting measurement noise without the use of any external filter the work first provides a framework for the design of the pid gains comprising of necessary and sufficient conditions for boundedness of trajectories and zero error convergence in presence of measurement noise it turns out that such convergence requires time varying gains subsequently novel recursive algorithms providing optimal and sub optimal time varying pid gains are proposed for discrete time varying linear multiple input multiple output mimo systems the development of the proposed optimal algorithm is based on minimising a stochastic performance index in presence of erroneous initial conditions white measurement noise and white process noise the proposed algorithms are shown to reject measurement noise provided that the system is asymptotically stable and the product of the input output coupling matrices is full column rank in addition convergence r

on the convergence of conservative difference schemes for the 2d generalized rosenau korteweg de vries equation abstract two conservative finite difference schemes for the rosenau kdv equation rkdv in 2d are proposed the first scheme is two level and nonlinear implicit the second scheme is three level and linear implicit existence of its difference solutions has been shown it is proved by the discrete energy method that the two schemes are uniquely solvable unconditionally stable and the convergence is of second order in the uniform norm numerical experiments demonstrate that the schemes are accurate and efficient

a new finite difference scheme for the rosenau burgers equation abstract in this paper a linear implicit finite difference scheme is given for the initial boundary problem of rosenau burgers equation which is convergent and unconditionally stable the unique solvability of numerical solutions has been shown a priori estimate and second order convergence of the finite difference approximate solution are discussed using energy method numerical results demonstrate that the scheme is efficient and accurate

a conservative fourth order stable finite difference scheme for the generalized rosenau kdv equation in both 1d and 2d abstract in the present work a conservative fourth order stable finite difference scheme is proposed to solve the generalized rosenau kdv equation in both 1d and 2d the existence uniqueness and mass and energy conservations of the numerical solution are proved using the discrete energy method the new scheme is convergent with o 2 h 4 and unconditionally stable numerical experiments are carried out to show that the scheme is efficient and reliable

an analytical numerical method for solving the korteweg de vries equation in this paper an analytical numerical method is applied to the one dimensional korteweg de vries equation with a variant of boundary and initial conditions to obtain its numerical solutions at small times two test problem with known exact solutions are studied to demonstrate the accuracy of the present method the obtained results are compared with the exact solution of each problem and are found to be in good agreement with each other the numerical scheme is also compared with earlier work and shown to be accurate and efficient

a fully conservative second order finite difference scheme for incompressible flow on nonuniform grids a second order accurate finite difference discretization of the incompressible navier stokes is presented that discretely conserves mass momentum and kinetic energy in the inviscid limit in space and time the method is thus completely free of numerical dissipation and potentially well suited to the direct numerical simulation or large eddy simulation of turbulent flow the method uses a staggered arrangement of velocity and pressure on a structured cartesian grid and retains its discrete conservation properties for both uniform and nonuniform gird spacing the predicted conservation properties are confirmed by inviscid simulations on both uniform and nonuniform grids the capability of the method to resolve turbulent flow is demonstrated by repeating the turbulent channel flow simulations of h choi and p moin 1994 j comput phys 113 1 where the effect of computational time step on the computed turbulence was investigated the present fully conservative scheme achieved turbulent flow solutions over the entire range of computational time steps investigated t tu2 0 4 to 5 0 little variation in statistical turbulence quantities was observed up to t 1 6 the present results differ significantly from those reported by choi and moin who observed significant discrepancies in the turbulence statistics above t 0 4 and the complete laminarization of the flow at and above t 1 6

explicit finite difference methods for the ew and rlw equations an extensive assessment of the accuracy of explicit finite difference methods for the solution of the equal width ew and regularized long wave rlw equations is reported such an assessment is based on the three invariants of these equations as well as on the magnitude of the errors of the numerical solution and has been performed as a function of the time step and grid spacing two of the methods presented here make use of three point fourth order accurate finite difference formulae for the first and second order spatial derivatives two methods are based on the analytical solution of second order ordinary differential equations which have locally exponential solutions and the fourth technique is a standard finite difference scheme a linear stability analysis of the four methods is presented it is shown that for the ew and rlw equations a compact operator method is more accurate than locally exponential techniques that make use of compact operator approximations the latter are reported to be more accurate than exponential techniques that employ second order accurate approximations and these in turn are more accurate than the standard explicit method

on the fibration method for zero cycles and rational points conjectures on the existence of zero cycles on arbitrary smooth projective varieties over number elds were proposed by colliot th el ene sansuc kato and saito in the 1980 s we prove that these conjectures are compatible with brations for brations into rationally connected varieties over a curve in particular they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers we prove the analogous result for rational points conditionally on a conjecture on locally split values of polynomials which a recent work of matthiesen establishes in the case of linear polynomials over the rationals

action of correspondences on filtrations on cohomology and 0 cycles of abelian varieties we prove that given a symmetrically distinguished correspondence of a suitable complex abelian variety which includes any abelian variety of dimension at most 5 powers of complex elliptic curves etc that vanishes as a morphism on a certain quotient of its middle singular cohomology then it vanishes as a morphism on the deepest part of a particular filtration on the chow group of 0 cycles of the abelian variety as a consequence we prove that an automorphism of such an abelian variety that acts as the identity on a certain quotient of its middle singular cohomology acts as the identity on the deepest part of this filtration on the chow group of 0 cycles of the abelian variety as an application we prove that for the generalized kummer variety associated to a complex abelian surface and the automorphism induced from a symplectic automorphism of the complex abelian surface the automorphism of the generalized kummer variety acts as the identity on a certain subgroup of its chow group of 0 cycles

on the blow up of solutions of the benjamin bona mahony burgers and rosenau burgers equations abstract in this paper we study sufficient conditions of the blow up of solutions of initial boundary value problems for the well known benjamin bona mahony burgers and rosenau burgers equations on a segment note that this is the first result for these equations in the blow up area

galerkin finite element method for the rosenau rlw equation finite element galerkin solutions for the rosenau rlw rrlw equation are considered the existence and uniqueness of the weak solution are shown and the semidiscrete galerkin scheme is studied using appropriate projections for a second order accuracy in time we propose the galerkin crank nicolson fully discrete method we also prove the convergence of the extrapolated crank nicolson scheme finally some numerical experiments are given to demonstrate the validity and accuracy of our method

an optimal stochastic multivariable pid controller a direct output tracking approach abstractthis paper deals with the design of an optimal stochastic controller possessing tracking capability of any reference output trajectory in the presence of measurement noise we consider multi input multi output linear time invariant systems and a proportional integral derivative pid controller the system under consideration needs not be stable a recursive algorithm providing optimal time varying pid gains is proposed for the case where the number of inputs is larger than or equal to the number of outputs the development of the proposed algorithm aims for per time sample minimisation of the mean square output error in the presence of erroneous initial conditions measurement noise and process noise necessary and sufficient conditions are provided for the convergence of the output error covariance in addition convergence results are presented for discretised continuous time plants simulation results are included to illustrate the performance capabilities of the proposed algorithm performan

a conservative linear difference scheme for the 2d regularized long wave equation abstract in the present work a three level in time linear and conservative implicit finite difference scheme for solving the 2d regularized long wave equation is proposed the existence uniqueness and conservations for mass and energy of the numerical solution are proved by the discrete energy method the new scheme is shown to be second order convergent and unconditionally stable numerical examples are provided to show the present scheme to be efficient and reliable

numerical method satisfying the first two conservation laws for the korteweg de vries equation in this paper we develop a finite volume scheme for the kdv equation which conserves both the momentum and energy the main ingredient of the method is a numerical device we developed in recent years that enables us to construct numerical method for a pde that also simulates its related equations in the method numerical approximations to both the momentum and energy are conservatively computed the operator splitting approach is adopted in constructing the method in which the conservation and dispersion parts of the equation are alternatively solved our numerical device is applied in solving the conservation part of the equation the feasibility and stability of the method is discussed which involves an important property of the method the so called jensen condition the truncation error of the method is analyzed which shows that the method is second order accurate finally several numerical examples including the zabusky kruskal s example are presented to show the good stability property of the method for long time numerical integration

bloch s conjecture for catanese and barlow surfaces catanese surfaces are regular surfaces of general type with pg 0 they specialize to double covers of barlow surfaces we prove that the ch0 group of a catanese surface is equal to z which implies the same result for the barlow surfaces

exponential finite difference method applied to korteweg de vries equation for small times the korteweg de vries equation is numerically solved by using the exponential finite difference technique the accuracy of computed solutions is examined by comparison with other numerical and analytical solutions using two examples the close results agreement between the current results and the exact solutions confirms that the proposed finite difference procedure is an effective technique for the solution of the korteweg de vries equation at the small times

a stochastic pid controller for a class of mimo systems abstractthis paper deals with the design of a controller possessing tracking capability of any realisable reference trajectory while rejecting measurement noise we consider discrete time varying multi input multi output stable linear systems and a proportional integral derivative pid controller a novel recursive algorithm estimating the time varying pid gains is proposed the development of the proposed algorithm is based on minimising a stochastic performance index the implementation of the proposed algorithm is described and boundedness of trajectories and convergence characteristics are presented for a discretised continuous time model simulation results are included to illustrate the performance capabilities of the proposed algorithm

a new finite difference scheme for generalized rosenau burgers equation in this paper the numerical solution of the initial boundary value problem of generalized rosenau burgers equation is considered a new linear implicit finite difference scheme of two level is proposed and the prior estimate of the finite difference solution is obtained the unique solvability of numerical solutions has been shown it is proved that the finite difference scheme is convergent and stable numerical experiments indicate the method is efficient

further advancements on the output feedback global continuous control for the finite time and exponential stabilisation of bounded input mechanical systems desired conservative force compensation and experiments global saturating proportional saturating derivative sp sd type continuous control for the finite time or local exponential stabilisation of mechanical systems with bounded inputs is achieved a

on tsertsvadze s difference scheme for the kuramoto tsuzuki equation in this paper we have proved the second order convergence in l norm of the tsertsvadze s difference scheme for the kuramoto tsuzuki equation the existence uniqueness and iterative algorithm are also discussed in detail furthermore a l second order convergent linearized difference scheme is given for inhomogeneous equation all results are obtained without any restrictions on the meshsizes at last a numerical example is presented

on the action of symplectic automorphisms on the mathrm ch _0 groups of some hyper kahler fourfolds we prove that for any polarized symplectic automorphism of the fano variety of lines of a smooth cubic fourfold equipped with the plucker polarization the induced action on the chow group of 0 cycles is identity as predicted by bloch beilinson conjecture we also prove the same result for the chow group of homologically trivial 2 cycles up to torsion

crank nicolson finite difference scheme for the rosenau burgers equation abstract in this paper a crank nicolson finite difference scheme for the numerical solution of the initial boundary value problem of rosenau burgers equation is proposed existence and uniqueness of numerical solutions are derived it is proved that the finite difference scheme is convergent in the order of o 2 h 2 and stable numerical simulations show that the method is efficient

on global output feedback regulation of euler lagrange systems with bounded inputs we identify a class of euler lagrange systems with bounded inputs that can be globally asymptotically stabilized via dynamic output feedback in particular we prove that if the system is fully actuated and the forces due to the potential field can be dominated by the constrained control signals then global output feedback regulation is still possible incorporating some suitable saturation functions in the controller

chow groups are finite dimensional in some sense when s is a surface with pg s 0 mumford proved that its chow group a s is not finite dimensional in some sense in this paper we propose another definition of finite dimensionality for the chow groups using this new definition at least the chow group of some surface s with pg s 0 for example the product of two curves becomes finite dimensional the finite dimensionality of the chow groups follows from the finite dimensionality of the chow motives it turns out that the finite dimensionality of the chow motives is a very strong property for example we can prove bloch s conjecture representability of the chow groups of surfaces with pg s 0 under the assumption that the chow motive of s is finite dimensional

brief semiglobal stability of saturated linear pid control for robot manipulators this paper addresses the position regulation problem of robot manipulators under control input constraints it is proven that the robot system under a saturated linear pid control is semiglobally asymptotically stable if the torque bounds are larger than gravitational torques and if the proportional gain is large enough and the integral gain is small enough the stability analysis makes use of singular perturbation tools and the results are illustrated via numerical simulations

numerical analysis for fourth order compact conservative difference scheme to solve the 3d rosenau rlw equation in this paper a fourth order compact and energy conservative difference scheme for three dimensional rosenau rlw equation is proposed the scheme is a two level and nonlinear implicit scheme it is proved by the discrete energy method that the compact scheme is solvable the convergence and stability of the difference scheme is obtained and its numerical convergence order is o 2 h4 in the l norm we discuss an iterative algorithm for solving the nonlinear algebraical system generated by the nonlinear compact scheme and prove its convergence numerical experiment results show that the theory is accurate and the method is efficient and reliable

efficient numerical schemes for fractional water wave models in this paper efficient numerical schemes are proposed for solving the fractional water wave models that describe the propagation of surface water wave by using the weighted and shifted grunwald letnikov wsgl formula to approximate the nonlocal fractional operators we design a series of second order accurate difference schemes for the considered models the existence stability and convergence of numerical solutions of the proposed numerical schemes are established rigorously the analysis shows that the proposed numerical schemes are unconditionally stable with second order accuracy for both temporal and spatial discretizations several numerical results are provided to verify the efficiency and accuracy of our theoretical analysis

asymptotic behavior of solutions to the rosenau burgers equation with a periodic initial boundary abstract this study focuses on the rosenau burgers equation u t u x x x x t u x x f u x 0 with a periodic initial boundary condition it is proved that with smooth initial value the global solution uniquely exists furthermore for 0 the global solution converges time asymptotically to the average of the initial value in an exponential form and the convergence rate is optimal while for 0 the unique solution oscillates around the initial average all the time finally the numerical simulations are reported to confirm the theoretical results

solitons and periodic solutions for the rosenau kdv and rosenau kawahara equations in this work we use the sine cosine and the tanh methods for solving the rosenau kdv and rosenau kawahara equations the two methods reveal solitons and periodic solutions the study confirms the power of the two schemes