A KdV-like advection-dispersion equation with some remarkable properties

We discuss a new non-linear PDE, u_t + (2 u_xx/u) u_x = epsilon u_xxx, invariant under scaling of dependent variable and referred to here as SIdV. It is one of the simplest such translation and space-time reflection-symmetric first order advection-dispersion equations. This PDE (with dispersion coefficient unity) was discovered in a genetic programming search for equations sharing the KdV solitary wave solution. It provides a bridge between non-linear advection, diffusion and dispersion. Special cases include the mKdV and linear dispersive equations. We identify two conservation laws, though initial investigations indicate that SIdV does not follow from a polynomial Lagrangian of the KdV sort. Nevertheless, it possesses solitary and periodic travelling waves. Moreover, numerical simulations reveal recurrence properties usually associated with integrable systems. KdV and SIdV are the simplest in an infinite dimensional family of equations sharing the KdV solitary wave. SIdV and its generalizations may serve as a testing ground for numerical and analytical techniques and be a rich source for further explorations.Comment: 15 pages, 4 figures, corrected sign typo in KdV Lagrangian above equation 3

Twist solitons in complex macromolecules: from DNA to polyethylene

DNA torsion dynamics is essential in the transcription process; simple models for it have been proposed by several authors, in particular Yakushevich (Y model). These are strongly related to models of DNA separation dynamics such as the one first proposed by Peyrard and Bishop (and developed by Dauxois, Barbi, Cocco and Monasson among others), but support topological solitons. We recently developed a ``composite'' version of the Y model, in which the sugar-phosphate group and the base are described by separate degrees of freedom. This at the same time fits experimental data better than the simple Y model, and shows dynamical phenomena, which are of interest beyond DNA dynamics. Of particular relevance are the mechanism for selecting the speed of solitons by tuning the physical parameters of the non linear medium and the hierarchal separation of the relevant degrees of freedom in ``master'' and ``slave''. These mechanisms apply not only do DNA, but also to more general macromolecules, as we show concretely by considering polyethylene.Comment: New version substantially longer, with new applications to Polyethylene. To appear in "International Journal of Non-Linear Mechanics

Modulational instability of two pairs of counter-propagating waves and energy exchange in two-component media

The dynamics of two pairs of counter-propagating waves in two-component media is considered within the framework of two generally nonintegrable coupled Sine-Gordon equations. We consider the dynamics of weakly nonlinear wave packets, and using an asymptotic multiple-scales expansion we obtain a suite of evolution equations to describe energy exchange between the two components of the system. Depending on the wave packet length-scale vis-a-vis the wave amplitude scale, these evolution equations are either four non-dispersive and nonlinearly coupled envelope equations, or four non-locally coupled nonlinear Schroedinger equations. We also consider a set of fully coupled nonlinear Schroedinger equations, even though this system contains small dispersive terms which are strictly beyond the leading order of the asymptotic multiple-scales expansion method. Using both the theoretical predictions following from these asymptotic models and numerical simulations of the original unapproximated equations, we investigate the stability of plane-wave solutions, and show that they may be modulationally unstable. These instabilities can then lead to the formation of localized structures, and to a modification of the energy exchange between the components. When the system is close to being integrable, the time-evolution is distinguished by a remarkable almost periodic sequence of energy exchange scenarios, with spatial patterns alternating between approximately uniform wavetrains and localized structures.Comment: 35 pages, 13 figure

Perturbed soliton excitations in DNA molecular chain

We study nonlinear dynamics of a periodic inhomogeneous DNA double helical chain under dynamic plane-base rotator model by considering angular rotation of bases in a plane normal to the helical axis. The dynamics is governed by a perturbed sine-Gordon equation. The perturbed soliton solution is obtained using a multiple scale soliton perturbation theory. The perturbed kink-antikink solitons represent formation of open state configuration with fluctuation in DNA.Comment: 20 Pages, 5 figure

Thermal diffusion of supersonic solitons in an anharmonic chain of atoms

We study the non-equilibrium diffusion dynamics of supersonic lattice solitons in a classical chain of atoms with nearest-neighbor interactions coupled to a heat bath. As a specific example we choose an interaction with cubic anharmonicity. The coupling between the system and a thermal bath with a given temperature is made by adding noise, delta-correlated in time and space, and damping to the set of discrete equations of motion. Working in the continuum limit and changing to the sound velocity frame we derive a Korteweg-de Vries equation with noise and damping. We apply a collective coordinate approach which yields two stochastic ODEs which are solved approximately by a perturbation analysis. This finally yields analytical expressions for the variances of the soliton position and velocity. We perform Langevin dynamics simulations for the original discrete system which fully confirm the predictions of our analytical calculations, namely noise-induced superdiffusive behavior which scales with the temperature and depends strongly on the initial soliton velocity. A normal diffusion behavior is observed for very low-energy solitons where the noise-induced phonons also make a significant contribution to the soliton diffusion.Comment: Submitted to PRE. Changes made: New simulations with a different method of soliton detection. The results and conclusions are not different from previous version. New appendixes containing information about the system energy and soliton profile

On the nonlinear dynamics of topological solitons in DNA

Dynamics of topological solitons describing open states in the DNA double helix are studied in the frameworks of the model which takes into account asymmetry of the helix. It is shown that three types of topological solitons can occur in the DNA double chain. Interaction between the solitons, their interactions with the chain inhomogeneities and stability of the solitons with respect to thermal oscillations are investigated.Comment: 16 pages, 16 figure

Some Recent Developments on Kink Collisions and Related Topics

We review recent works on modeling of dynamics of kinks in 1+1 dimensional $\phi^4$ theory and other related models, like sine-Gordon model or $\phi^6$ theory. We discuss how the spectral structure of small perturbations can affect the dynamics of non-perturbative states, such as kinks or oscillons. We describe different mechanisms, which may lead to the occurrence of the resonant structure in the kink-antikink collisions. We explain the origin of the radiation pressure mechanism, in particular, the appearance of the negative radiation pressure in the $\phi^4$ and $\phi^6$ models. We also show that the process of production of the kink-antikink pairs, induced by radiation is chaotic.Comment: 26 pages, 9 figures; invited chapter to "A dynamical perspective on the {\phi}4 model: Past, present and future", Eds. P.G. Kevrekidis and J. Cuevas-Maraver; Springer book class with svmult.cls include