Analytical and numerical studies of the variable-coefficient Gardner equation

The Gardner or extended Korteweg-de Vries equation with the variable-sign cubic non-linear term is studied. This equation occurred in the theory of interfacial waves in the stratified shear flow at some certain conditions on the medium stratification. An asymptotic method is applied to study the solitary wave transformation when the coefficient of the cubic non-linear term is varied slowly. It is found the one branch of the solitary waves destroys in the vicinity of the bifurcation point, where the exponential soliton transforms into the breather through algebraic soliton. Other solitary waves change their amplitude accordingly to the cubic non-linear term variation. The structure of the tail formed behind the solitary wave is determined. Direct numerical simulation of the Gardner equation is performed to analyze the transformation and breaking of the solitary wave. It confirms most of the theoretical results. The calculated dynamics of the ''table-top'' soliton due to the large values of the mass and momentum integrals differs from the asymptotic prediction because the characteristic time of the variation of the cubic non-linear term has the same order as the duration of ''table-top'' soliton.

[1]  Efim Pelinovsky,et al.  Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface , 2002 .

[2]  R. Grimshaw,et al.  Solitary Wave Transformation Due to a Change in Polarity , 1998 .

[3]  J. Miles.  On internal solitary waves. II , 1981 .

[4]  R. Grimshaw,et al.  Structural transformation of eigenvalues for a perturbed algebraic soliton potential , 1997 .

[5]  Efim Pelinovsky,et al.  Internal Tide Transformation and Oceanic Internal Solitary Waves , 2003 .

[6]  S. Novikov,et al.  Theory of Solitons: The Inverse Scattering Method , 1984 .

[7]  K. Lamb,et al.  The Evolution of Internal Wave Undular Bores: Comparisons of a Fully Nonlinear Numerical Model with Weakly Nonlinear Theory , 1996 .

[8]  R. Grimshaw.  Internal Solitary Waves , 2003 .

[9]  Efim Pelinovsky,et al.  Solitary wave transformation in a medium with sign-variable quadratic nonlinearity and cubic nonlinearity , 1999 .

[10]  P. Clarkson,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering: References , 1991 .

[11]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[12]  A. Slyunyaev,et al.  Dynamics of large-amplitude solitons , 1999 .

[13]  R. Grimshaw,et al.  Generation of large-amplitude solitons in the extended Korteweg-de Vries equation. , 2002, Chaos.

[14]  Soliton creation and destruction, resonant interactions, and inelastic collisions in shallow water waves , 1998 .

[15]  Electromagnetic compensating gauge transformations , 1971 .

[16]  Yuji Kodama,et al.  Perturbations of solitons and solitary waves , 1981 .

[17]  Alan C. Newell,et al.  Solitons in mathematics and physics , 1987 .

[18]  Yuri S. Kivshar,et al.  Dynamics of Solitons in Nearly Integrable Systems , 1989 .

[19]  A. Slyunyaev.  Dynamics of localized waves with large amplitude in a weakly dispersive medium with a quadratic and positive cubic nonlinearity , 2001 .

[20]  E. Pelinovsky,et al.  Breaking of solitary and periodic nonlinear waves , 1981 .

[21]  P. Drazin,et al.  Solitons: An Introduction , 1989 .

[22]  A. Newell,et al.  Internal solitary waves near a turning point , 1980 .

[23]  V. Karpman,et al.  Soliton Evolution in the Presence of Perturbation , 1979 .

[24]  B. Malomed,et al.  Soliton caustics , 1991 .

[25]  T. Kakutani,et al.  Solitary Waves on a Two-Layer Fluid , 1978 .

[26]  L. Ostrovsky,et al.  Perturbation Theories for Nonlinear Waves , 2000 .

[27]  Timothy R. Marchant,et al.  Soliton interaction for the extended Korteweg-de Vries equation , 1996 .

[28]  Lev A. Ostrovsky,et al.  Interactions of solitons in nonintegrable systems: Direct perturbation method and applications , 1981 .

[29]  R. Grimshaw,et al.  Interaction of a solitary wave with an external force , 1994 .

[30]  R. Grimshaw,et al.  A second-order theory for solitary waves in shallow fluids , 1983 .

[31]  Efim Pelinovsky,et al.  A generalized Korteweg-de Vries model of internal tide transformation in the coastal zone , 1999 .