Multisymplectic implicit and explicit methods for Klein-Gordon-Schrdinger equations

We propose multisymplectic implicit and explicit Fourier pseudospectral methods for the Klein-Gordon-Schrodinger equations.We prove that the implicit method satisfies the charge conservation law exactly.Both methods provide accurate solutions in long-time computations and simulate the soliton collision well.The numerical results show the abilities of the two methods in preserving the charge,energy,and momentum conservation laws.

A new explicit multisymplectic integrator for the Kawahara-type equation

We derive a new multisymplectic integrator for the Kawahara-type equation which is a fully explicit scheme and thus needs less computation cost. Multisympecticity of such scheme guarantees the long-time numerical behaviors. Nu- merical experiments are presented to verify the accuracy of this scheme as well as the excellent performance on invariant preservation for three kinds of Kawahara-type equations.

Concatenating construction of the multisymplectic schemes for 2+1-dimensional sine-Gordon equation

In this paper, taking the 2+1-dimensional sine-Gordon equation as an example, we present the concatenating method to construct the multisymplectic schemes. The method is to discretizee independently the PDEs in different directions with symplectic schemes, so that the multisymplectic schemes can be constructed by concatenating those symplectic schemes. By this method, we can construct multisymplectic schemes, including some widely used schemes with an accuracy of any order. The numerical simulation on the collisions of solitons are also proposed to illustrate the efficiency of the multisymplectic schemes.

MULTISYMPLECTIC FOURIER PSEUDOSPECTRAL METHOD FOR THE NONLINEAR SCHR■DINGER EQUATIONS WITH WAVE OPERATOR

In this paper, the multisymplectic Fourier pseudospectral scheme for initial-boundary value problems of nonlinear SchrSdinger equations with wave operator is considered. We investigate the local and global conservation properties of the multisymplectic discretization based on Fourier pseudospectral approximations. The local and global spatial conservation of energy is proved. The error estimates of local energy conservation law are also derived. Numerical experiments are presented to verify the theoretical predications.

Multisymplectic five-point scheme for the nonlinear wave equation

In this paper, we introduce the multisymplecticstructure of the nonlinear wave equation, and prove that theclassical five-point scheme for the equation is multisymplec-tic. Numerical simulations of this multisymplectic scheme onhighly oscillatory waves of the nonlinear Klein-Gordonequation and the collisions between kink and anti-kink soli-tons of the sine-Gordon equation are also provided. The mul-tisymplectic schemes do not need to discrete PDEs in thespace first as the symplectic schemes do and preserve notonly the geometric structure of the PDEs accurately, but alsotheir first integrals approximately such as the energy, themomentum and so on. Thus the multisymplectic schemeshave better numerical stability and long-time numerical be-havior than the energy-conserving scheme and the symplec-tic scheme.

Multisymplectic Structure and Multisymplectic Scheme for the Nonlinear Ware Equation

The multisymplectic structure of the nonlinear wave equation is derived directly from the variational principle. In the numerical aspect, we present a multisymplectic nine points scheme which is equivalent to the multisymplectic Preissman scheme. A series of numerical results are reported to illustrate the effectiveness of the scheme.

HIGH ORDER COMPACT MULTISYMPLECTIC SCHEME FOR COUPLED NONLINEAR SCHRODINGER-KDV EQUATIONS

In this paper, a novel multisymplectic scheme is proposed for the coupled nonlinear Schrodinger-KdV （CNLS-KdV） equations. The CNLS-KdV equations are rewritten into the multisymplectic Hamiltonian form by introducing some canonical momenta. To simulate the problem efficiently, the CNLS-KdV equations are approximated by a high order compact method in space which preserves N semi-discrete multisymplectic conservation laws. We then discretize the semi-discrete system by using a symplectic midpoint scheme in time. Thus, a full-discrete multisymplectic scheme is obtained for the CNLS-KdV equations. The conservation laws of the full-discrete scheme are analyzed. Some numerical experiments are presented to further verify the convergence and conservation laws of the new scheme.

MULTISYMPLECTIC COMPOSITION INTEGRATORS OF HIGH ORDER

A composition method for constructing high order multisymplectic integrators is presented in this paper. The basic idea is to apply composition method to both the time and the space directions. We also obtain a general formula for composition method.

Interaction effects of DNA, RNA-polymerase, and cellular fluid on the local dynamic behaviors of DNA

In view of the complex structure and environment,the dynamic analysis on deoxyribonucleic acid(DNA)is a challenge in the biophysics field.Considering the local interaction with ribonucleic acid(RNA)-polymerase as well as the dissipative effect of cellular fluid,a coupling sine-Gordon-type dynamic model is used to describe the rotational motions of the bases in DNA.First,the approximate symmetric form is constructed.Then,the wave form and the wave velocity of the kink solution to the proposed dynamic model are investigated by a Runge-Kutta structure-preserving scheme based on the generalized multi-symplectic idea.The numerical results indicate that,the strengthening of the local interaction between DNA and RNA-polymerase described by the coupling potential makes the form of the kink solution steep,while the appearance of the friction between DNA and cellular fluid makes the form of the kink solution flat.In addition,the appearance of the friction decreases the velocities of both the symplectic configuration and the anti-symplectic configuration with different degrees.The above findings are beneficial to comprehend the DNA transcription mechanism.

Effects of temperature change on the rheological property of modified multiwall carbon nanotubes

Solvent-free nanofluids hold promise for many technologically significant applications.The liquid-like behavior,a typical rheological property of solvent-free nanofluids,has aroused considerable interests.However,there has been still lack of efficient methods to predict and control the liquid-like behavior of solvent-free nanofluids.In this paper,we propose a semi-discrete dynamic system with stochastic excitation describing the temperature change effects on the rheological property of multiwall carbon nanotubes(MWCNTs)modified by grafting sulfonic acid terminated organosilanes as corona and tertiary amine as canopy,which is a typical covalent-type solvent-free nanofluid system.The vibration of the grafting branches is simulated by employing a structure-preserving approach,and the shear force of grafting branches at the fixed end is computed subsequently.By taking the shear forces as an excitation acting on the MWCNTs,the axial motion of the MWCNTs is solved with the 7-point Gauss-Kronrod quadrature rule.The critical temperature associated with the appearance of the liquid-like behavior as well as the upper bound of the moving speed of the modified MWCNTs is determined,which can be used to predict and control the liquid-like behavior of the modified MWCNTs in engineering applications.

GeometricNumerical Integration for Peakon b-Family Equations

In this paper,we study the Camassa-Holm equation and the Degasperis-Procesi equation.The two equations are in the family of integrable peakon equations,and both have very rich geometric properties.Based on these geometric structures,we construct the geometric numerical integrators for simulating their soliton solutions.The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties,however they also have the significant difference,for example there exist the shock wave solutions for the Degasperis-Procesi equation.By using the symplectic Fourier pseudo-spectral integrator,we simulate the peakon solutions of the two equations.To illustrate the smooth solitons and shock wave solutions of the DP equation,we use the splitting technique and combine the composition methods.In the numerical experiments,comparisons of these two kinds of methods are presented in terms of accuracy,computational cost and invariants preservation.