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

Thermal conductivity of one-dimensional lattices with self-consistent heat baths: a heuristic derivation

We derive the thermal conductivities of one-dimensional harmonic and anharmonic lattices with self-consistent heat baths (BRV lattice) from the Single-Mode Relaxation Time (SMRT) approximation. For harmonic lattice, we obtain the same result as previous works. However, our approach is heuristic and reveals phonon picture explicitly within the heat transport process. The results for harmonic and anharmonic lattices are compared with numerical calculations from Green-Kubo formula. The consistency between derivation and simulation strongly supports that effective (renormalized) phonons are energy carriers in anharmonic lattices although there exist some other excitations such as solitons and breathers.Comment: 4 pages, 3 figures. accepted for publication in JPS

q-Breathers and thermalization in acoustic chains with arbitrary nonlinearity index

Nonlinearity shapes lattice dynamics affecting vibrational spectrum, transport and thermalization phenomena. Beside breathers and solitons one finds the third fundamental class of nonlinear modes -- $q$-breathers -- periodic orbits in nonlinear lattices, exponentially localized in the reciprocal mode space. To date, the studies of $q$-breathers have been confined to the cubic and quartic nonlinearity in the interaction potential. In this paper we study the case of arbitrary nonlinearity index $\gamma$ in an acoustic chain. We uncover qualitative difference in the scaling of delocalization and stability thresholds of $q$-breathers with the system size: there exists a critical index $\gamma^*=6$, below which both thresholds (in nonlinearity strength) tend to zero, and diverge when above. We also demonstrate that this critical index value is decisive for the presence or absense of thermalization. For a generic interaction potential the mode space localized dynamics is determined only by the three lowest order nonlinear terms in the power series expansion.Comment: 5 pages, 4 figure

Solitons on the edge of a two-dimensional electron system

We present a study of the excitations of the edge of a two-dimensional electron droplet in a magnetic field in terms of a contour dynamics formalism. We find that, beyond the usual linear approximation, the non-linear analysis yields soliton solutions which correspond to uniformly rotating shapes. These modes are found from a perturbative treatment of a non-linear eigenvalue problem, and as solutions to a modified Korteweg-de Vries equation resulting from a local induction approximation to the nonlocal contour dynamics. We discuss applications to the edge modes in the quantum Hall effect.Comment: 4 pages, 2 eps figures (included); to appear in Phys. Rev. Letter

Discrete Multiscale Analysis: A Biatomic Lattice System

We discuss a discrete approach to the multiscale reductive perturbative method and apply it to a biatomic chain with a nonlinear interaction between the atoms. This system is important to describe the time evolution of localized solitonic excitations. We require that also the reduced equation be discrete. To do so coherently we need to discretize the time variable to be able to get asymptotic discrete waves and carry out a discrete multiscale expansion around them. Our resulting nonlinear equation will be a kind of discrete Nonlinear Schr\"odinger equation. If we make its continuum limit, we obtain the standard Nonlinear Schr\"odinger differential equation

Physics on the edge: contour dynamics, waves and solitons in the quantum Hall effect

We present a theoretical study of the excitations on the edge of a two-dimensional electron system in a perpendicular magnetic field in terms of a contour dynamics formalism. In particular, we focus on edge excitations in the quantum Hall effect. Beyond the usual linear approximation, a non-linear analysis of the shape deformations of an incompressible droplet yields soliton solutions which correspond to shapes that propagate without distortion. A perturbative analysis is used and the results are compared to analogous systems, like vortex patches in ideal hydrodynamics. Under a local induction approximation we find that the contour dynamics is described by a non-linear partial differential equation for the curvature: the modified Korteweg-de Vries equation. PACS number(s): 73.40.Hm, 02.40.Ma, 03.40.Gc, 11.10.LmComment: 15 pages, 12 embedded figures, submitted to Phys. Rev.

The Fermi-Pasta-Ulam recurrence and related phenomena for 1D shallow-water waves in a finite basin

In this work, different regimes of the Fermi-Pasta-Ulam (FPU) recurrence are simulated numerically for fully nonlinear "one-dimensional" potential water waves in a finite-depth flume between two vertical walls. In such systems, the FPU recurrence is closely related to the dynamics of coherent structures approximately corresponding to solitons of the integrable Boussinesq system. A simplest periodic solution of the Boussinesq model, describing a single soliton between the walls, is presented in an analytical form in terms of the elliptic Jacobi functions. In the numerical experiments, it is observed that depending on a number of solitons in the flume and their parameters, the FPU recurrence can occur in a simple or complicated manner, or be practically absent. For comparison, the nonlinear dynamics of potential water waves over nonuniform beds is simulated, with initial states taken in the form of several pairs of colliding solitons. With a mild-slope bed profile, a typical phenomenon in the course of evolution is appearance of relatively high (rogue) waves, while for random, relatively short-correlated bed profiles it is either appearance of tall waves, or formation of sharp crests at moderate-height waves.Comment: revtex4, 10 pages, 33 figure

Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates

Solitons are among the most distinguishing fundamental excitations in a wide range of non-linear systems such as water in narrow channels, high speed optical communication, molecular biology and astrophysics. Stabilized by a balance between spreading and focusing, solitons are wavepackets, which share some exceptional generic features like form-stability and particle-like properties. Ultra-cold quantum gases represent very pure and well-controlled non-linear systems, therefore offering unique possibilities to study soliton dynamics. Here we report on the first observation of long-lived dark and dark-bright solitons with lifetimes of up to several seconds as well as their dynamics in highly stable optically trapped $^{87}$Rb Bose-Einstein condensates. In particular, our detailed studies of dark and dark-bright soliton oscillations reveal the particle-like nature of these collective excitations for the first time. In addition, we discuss the collision between these two types of solitary excitations in Bose-Einstein condensates.Comment: 9 pages, 4 figure

Asymmetric gap soliton modes in diatomic lattices with cubic and quartic nonlinearity

Nonlinear localized excitations in one-dimensional diatomic lattices with cubic and quartic nonlinearity are considered analytically by a quasi-discreteness approach. The criteria for the occurence of asymmetric gap solitons (with vibrating frequency lying in the gap of phonon bands) and small-amplitude, asymmetric intrinsic localized modes (with the vibrating frequency being above all the phonon bands) are obtained explicitly based on the modulational instabilities of corresponding linear lattice plane waves. The expressions of particle displacement for all these nonlinear localized excitations are also given. The result is applied to standard two-body potentials of the Toda, Born-Mayer-Coulomb, Lennard-Jones, and Morse type. The comparison with previous numerical study of the anharmonic gap modes in diatomic lattices for the standard two-body potentials is made and good agreement is found.Comment: 24 pages in Revtex, 2 PS figure