Competition between geometric dispersion and viscous dissipation in wave propagation of KdV-Burgers equation

Abstract

In this paper the competitive relationship between the geometric dispersion and the viscous dissipation in the wave propagation of the KdV-Burgers equation is investigated by the generalized multi-symplectic method. Firstly, the generalized multi-symplectic formulations for the KdV-Burgers equation are presented in Hamiltonian space. Then, focusing on the inherent geometric properties of the generalized multi-symplectic formulations, a 12-point difference scheme is constructed. Finally, numerical experiments are performed with fixed step-sizes to obtain the maximum damping coefficient that insures that the scheme constructed is generalized multi-symplectic, and to study the competition between the geometric dispersion and the viscous dissipation in the wave propagation of the KdV-Burgers equation. The competition phenomena are comprehensively illustrated in the wave forms as well as in the phase diagrams: for the KdV equation (a particular case of the KdV-Burgers equation without dissipation), there is a closed orbit in the phase diagram; and the closed orbit is substituted by a heteroclinic one with the appearance of the viscous dissipation; moreover, the heteroclinic orbit changes from the saddle-node type to the saddle-focus type with an increase of the damping coefficient.

Almost structure-preserving analysis for weakly linear damping nonlinear Schrödinger equation with periodic perturbation

“…Form Eq. 12, it can be found that the form of the generalized multi-symplectic conservation law perturbation is similar to those presented in our previous jobs [43,45] even if the damping rate is not contained in the coefficient matrices of Eq. (4), but in ( , , , ) p q x t ε .…”

“…Different from the generalized multi-symplectic forms those we presented for the damping systems considered in Refs. [43][44][45][46], the linear damping rate isn't contained in the coefficient matrices of Eq. (4), but in the addition term ( , , , ) p q x t ε .…”

Complementary wave solutions for the long-short wave resonance model via the extended trial equation method and the generalized Kudryashov method

“…We have been able to derive wave solution of kink-type (40), dark soliton solution of the kink-type for S and the bell-type for L (42) as well as a compound wave solution of the bell-type and the kink-type for S and L (44).…”

“…They o er both a concise and an e cient way for solving NPDEs, in two or more dimensions. Other methods, such as multi-symplectic method [37,38], or general-ized multi-symplectic method [39,40] will be employed in future works.…”

Lie symmetry analysis and conservation laws for the time fractional simplified modified Kawahara equation

“…(1) reduces to the classical SMK equation which was considered for exact travelling wave solutions and Cls in [49][50][51]. Moreover, one can nd more details on the construction of analytical, exact, numerical solutions, and other information for classical NLPDEs, in [52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69].…”