new conservative schemes with discrete variational derivatives for nonlinear wave equations new conservative finite difference schemes for certain classes of nonlinear wave equations are proposed the key tool there is discrete variational derivative by which discrete conservation property is realized a similar approach for the target equations was recently proposed by furihata but in this paper a different approach is explored where the target equations are first transformed to the equivalent system representations which are more natural forms to see conservation properties applications for the nonlinear klein gordon equation and the so called good boussinesq equation are presented numerical examples reveal the good performance of the new schemes

symplectic and multisymplectic lobatto methods for the good boussinesq equation in this paper we construct second order symplectic and multisymplectic integrators for the good boussineq equation using the two stage lobatto iiia iiib partitioned runge kutta method which yield an explicit scheme and is equivalent to the classical central difference approximation to the second order spatial derivative numerical dispersion properties and the stability of both integrators are investigated numerical results for different solitary wave solutions confirm the excellent long time behavior of symplectic and multisymplectic integrators by preserving local and global energy and momentum

nonlinear stability and convergence of finite difference methods for the good boussinesq equation the good boussinesq equationu tt u xxxx u xx u 2 xx has recently been found to possess an interesting soliton interaction mechanism in this paper we study the nonlinear stability and the convergence of some simple finite difference schemes for the numerical solution of problems involving the good boussinesq equation numerical experimentas are also reported

multisymplectic geometry local conservation laws and fourier pseudospectral discretization for the good boussinesq equation the multisymplectic geometry for the good boussinesq equation is presented in this paper the multisymplectic form and the local energy and momentum conservation laws are derived directly from the variational principle the multisymplectic hamiltonian formulation is also presented based on the multisymplectic formulation we investigate the corresponding multisymplectic fourier pseudospectral discretization

soliton and antisoliton interactions in the good boussinesq equation the solitary wave interaction mechanism for the good boussinesq equation is investigated and found to be far more complicated than was previously thought three salient features are that solitary waves only exist for a finite range of velocities that large solitons can turn into so called antisolitons and that it is possible for solitons to merge and split small solitons however appear to be stable the existence of a potential well is linked to the different behaviors observed between small and large initial conditions

blow up and scattering of solution for a generalized boussinesq equation in this paper we consider the cauchy problem for a class of boussinesq equation we obtain the existence and uniqueness of the local solutions for a class of nonlinearity of the perturbation blow up solutions are obtained furthermore the global existence and nonlinear scattering for small amplitude solutions are established

multisoliton solution of a class of coupled soliton equations abstract the n soliton solutions of the korteweg devries kdv and boussinesq equations with a source are obtained explicitly by using the bilinear transformation method also the interactions of two soliton solutions are studied and the explicit forms for the one and two soliton solutions are obtained for the two systems

on certain homological invariants of groups recently the notions of sfli the supremum of the flat lengths of injective modules and silf the supremum of the injective lengths of flat modules have been studied by some authors these homological invariants are based on spli and silp invariants of gedrich and gruenberg and it is shown that they have enough potential to play an important role in studying homological conjectures in cohomology of groups in this paper we will study these invariants it turns out that for any group the finiteness of silf implies the finiteness of sfli but the converse is not known we investigate the situation in which sfli implies silf the statement holds for example for groups with the property that flat modules have finite projective dimension moreover we show that the gorenstein flat dimension of the trivial z module z that will be called gorenstein homological dimension of denoted ghd is completely related to these invariants

finite element galerkin method for the good boussinesq equation recently it has been shown by manoranjan et al l that this class of equations possesses an interesting solitary wave interaction mechanism moreover using semi group theoretic approch they have established global existence of a generalized solution provided the initial data are in a potential well further it is shown that finite time blowup may occur for weak solution in r if the initial data are not in the potential well and the total energy at t 0 exceeds the depth of the well in the first part of this paper we shall discuss existence uniqueness and regularity results for the problem 1 i 1 3 using the faedo galerkin method in the literature very few results are available pertaining to the numerical study of 1 l 1 4 in 2 defrutos et al have analysed pseudo spectral fourier method for the equation i i i 2 with periodic boundary conditions they have also discussed the nonlinear stability and convergence results in the analytic framework introduced earlier by lopez marcos and sanz serna 3 for related results on finite difference schemes one may refer to ortega and sanz serna 4 recently manoranjan et al 5 have applied petrov galerkin finite element method for cauchy problem without rigorous error analysis and have presented some numerical results in the later part of the present paper we have derived optima1 error estimates for both semidiscrete

highly accurate compact difference scheme for fourth order parabolic equation with dirichlet and neumann boundary conditions application to good boussinesq equation abstract in this work a three level implicit compact difference scheme for the generalised form of fourth order parabolic partial differential equation is developed the discretization is derived by approximating the lower order derivative terms using the governing differential equation with the imbedding technique and is fourth order accurate in space and second order accurate in time the current approach is advantageous since the boundary conditions are completely satisfied and no further approximations are required to be carried out at the boundaries the ability of the proposed scheme in handling linear singular problems is examined the value of first order space derivative is computed alongwith the solution so it does not have to be estimated using the calculated value of the solution the method successfully works for the highly nonlinear good boussinesq equation for which more accurate solutions are obtained for the single and the double soliton solutions in comparison with the existing numerical methods

the wahlquist estabrook method for evolution equations with a small parameter a technique of approximating pseudopotentials an approach is suggested to construct approximate solutions for nonlinear partial differential equations non integrable by the inverse scattering transform the method is applied to find an approximate lax pair and conservation laws for the kawahara equation

numerical solutions for the damped boussinesq equation by fd fft perturbation method abstract unconditional stability and consistency are considered using an implicit finite difference finite fourier transform perturbation technique for the damped boussinesq equation convergence and error analysis for the difference scheme are also discussed

on a direct bilinearization method kaup s higher order water wave equation as a modified nonlocal boussinesq equation a systematic procedure for the bilinearization of classes of soliton equations is developed with the help of a generalization of bell s exponential polynomials 1934 application of this procedure to kaup s higher order wave equation 1975 discloses several links with other soliton systems in particular it is found that the kaup equation is the modified version of a sech square soliton system which constitutes an alternative to the good boussinesq equation

derivation of lagrangian density for the good boussinesq equation and multisymplectic disretizations abstract the lagrangian density for the good boussinesq equation is derived by introducing a new variable in this paper based on this density we consider the corresponding multisymplectic formulation and discretizations numerical experiments are also reported

decay and scattering of solutions for a generalized boussinesq equation abstract in this paper we consider the long time behavior of small solutions of the cauchy problem for a generalized boussinesq equation a scattering operator and the nonlinear scattering for small amplitude solutions of the boussinesq equation are established under certain hypotheses

two exponential type integrators for the good boussinesq equation we introduce two exponential type integrators for the good boussinesq equation they are of orders one and two respectively and they require lower spatial regularity of the solution compared to classical exponential integrators for the first integrator we prove first order convergence in h r for solutions in h r 1 with r 1 2 this new integrator even converges with lower order in h r for solutions in h r i e without any additional smoothness assumptions for the second integrator we prove second order convergence in h r for solutions in h r 3 with r 1 2 and convergence in l 2 for solutions in h 3 numerical results are reported to illustrate the established error estimates the experiments clearly demonstrate that the new exponential type integrators are favorable over classical exponential integrators for initial data with low regularity

an integral homological characterization of finite groups abstract we show that a group g is finite if and only if every injective z g module has projective length one

rigorous numerics for ill posed pdes periodic orbits in the boussinesq equation in this paper we develop computer assisted techniques for the analysis of periodic orbits of ill posed partial differential equations as a case study our proposed method is applied to the boussinesq equation which has been investigated extensively because of its role in the theory of shallow water waves the idea is to use the symmetry of the solutions and a newton kantorovich type argument the radii polynomial approach to obtain rigorous proofs of existence of the periodic orbits in a weighted l1 banach space of space time fourier coefficients with exponential decay we present several computer assisted proofs of the existence of periodic orbits at different parameter values

on certain cohomological invariants of groups abstract let r be a left and right 0 noetherian ring we show that if all projective left and all projective right r modules have finite injective dimension then all injective left and all injective right r modules have finite projective dimension using this result we prove that the invariants silp z g and spli z g which were introduced by gedrich and gruenberg 1987 15 are equal for any group g as an application of the latter equality we show that a group g is finite if and only if cd g 0 where cd is the generalized cohomological dimension of groups introduced by ikenaga 1984 21

decay estimates for a dissipative dispersive linear semigroup and application to the viscous boussinesq equation abstract at the core of this article is the new estimate for a class of dissipative dispersive linear semigroup e t i p t arising in the study of the viscous boussinesq equation we combine the decay estimate with introducing a set of time weighted sobolev spaces where the time weights and the regularity of the sobolev spaces are determined by our decay estimate to show the global existence and asymptotic behavior of solutions to the viscous boussinesq equation in r n

pseudospectral method for the good boussinesq equation we prove the nonlinear stability and convergence of a fully discrete pseudospectral scheme for the good boussinesq equation utt uxxxx uxx u2 xx numerical comparisons with finite difference schemes are also reported

a homological characterization of locally finite groups abstract we show that a group g is locally finite if and only if tor 1 z g m i 0 for all z g modules m and i with m z free and i injective to that end we relate the finiteness of a finitely generated group g to the finiteness of the injective dimension of induced modules we also examine the relation between the finiteness of a group g and the finiteness of the projective dimension of coinduced modules