Toda Lattice Model Research Articles

Chaotic string motion in a near pp-wave limit

We revisit classical string motion in a near pp-wave limit of AdS5 × S5. It is known that the Toda lattice models are integrable. But if the exponential potential is truncated at finite order, then the system may become non-integrable. In particular, when the exponential potential in a three-particle periodic Toda chain is truncated at the third order of the dynamical variables, the resulting system becomes a well-known non-integrable system, Henon-Heiles model. The same thing may happen in a near pp-wave limit of AdS5 × S5, on which the classical string motion becomes chaotic.

Head-on Collision of Solitary Waves Described by the Toda Lattice Model in Granular ChainSupported by the National Natural Science Foundation of China (Grant No. 11574153).

We study the head-on collision of two solitary waves in a precompressed granular chain using the discrete element method. Our study takes the Toda chain solution as the initial condition for the simulations. The simulation covers the dynamical evolution of the collision process from the start of the incident wave to the end of the collision. The interaction has a central collision region of about five-grain width in which two solitary waves merge completely and share only one peak. Four stages, i.e., the pre-in-phase traveling stage, lag-phase collision state, lead-phase collision state, and post-in-phase traveling stage, are identified to describe the complex collision processes. Our results may be helpful for explaining the existence of long-lived solitary waves seen in the simulations by Takato and Sen [Europhys. Lett. 100 (2012) 24003].

Kinetic theory of quantum and classical Toda lattices

We consider the kinetic theory of the quantum and classical Toda lattice models. A kinetic equation of Bethe–Boltzmann type is derived for the distribution function of conserved quasiparticles. Near the classical limit, we show that the kinetic theory depends smoothly on Planck’s constant, and explicitly characterise the leading quantum corrections to classical behaviour. The classical kinetic theory is compared with direct numerical simulations and shows excellent agreement. Finally, we connect the Bethe–Boltzmann approach with the classical inverse scattering method, by identifying conserved quasiparticles with the spectrum of the Lax matrix.

A bound on quantum chaos from Random Matrix Theory with Gaussian Unitary Ensemble

In this article, using the principles of Random Matrix Theory (RMT) with Gaussian Unitary Ensemble (GUE), we give a measure of quantum chaos by quantifying Spectral From Factor (SFF) appearing from the computation of two point Out of Time Order Correlation function (OTOC) expressed in terms of square of the commutator bracket of quantum operators which are separated in time scale. We also provide a strict model independent bound on the measure of quantum chaos, −1/N (1 − 1/π) ≤ SFF ≤ 0 and 0 ≤ SFF ≤ 1/πN, valid for thermal systems with large and small number of degrees of freedom respectively. We have studied both the early and late behaviour of SFF to check the validity and applicability of our derived bound. Based on the appropriate physical arguments we give a precise mathematical derivation to establish this alternative strict bound of quantum chaos. Finally, we provide an example of integrability from GUE based RMT from Toda Lattice model to explicitly show the application of our derived bound on SFF to quantify chaos.

Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral

We investigate the long term evolution of trajectories in the Fermi-Pasta-Ulam (FPU) system, using as a probe the first non-trivial integral $J$ in the hierarchy of integrals of the corresponding Toda lattice model. To this end we perform simulations of FPU-trajectories for various classes of initial conditions produced by the excitation of isolated modes, packets, as well as 'generic' (random) initial data. For initial conditions corresponding to localized energy excitations, $J$ exhibits variations yielding 'sigmoid' curves similar to observables used in literature, e.g., the 'spectral entropy' or various types of 'correlation functions'. However, $J(t)$ is free of fluctuations inherent in such observables, hence it constitutes an ideal observable for probing the timescales involved in the stages of FPU dynamics. We observe two fundamental timescales: i) the 'time of stability' (in which, roughly, FPU trajectories behave like Toda), and ii) the 'time to equilibrium' (beyond which energy equipartition is reached). Below a specific energy crossover, both times are found to scale exponentially as an inverse power of the specific energy. However, this crossover goes to zero with increasing the degrees of freedom $N$ as $\varepsilon _c \sim N^{-b}$, with $b \in [1.5, 2.5]$. For 'generic data' initial conditions, instead, $J(t)$ allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori.

Solitons riding on solitons and the quantum Newton's cradle.

The reduced dynamics for dark and bright soliton chains in the one-dimensional nonlinear Schrödinger equation is used to study the behavior of collective compression waves corresponding to Toda lattice solitons. We coin the term hypersoliton to describe such solitary waves riding on a chain of solitons. It is observed that in the case of dark soliton chains, the formulated reduction dynamics provides an accurate an robust evolution of traveling hypersolitons. As an application to Bose-Einstein condensates trapped in a standard harmonic potential, we study the case of a finite dark soliton chain confined at the center of the trap. When the central chain is hit by a dark soliton, the energy is transferred through the chain as a hypersoliton that, in turn, ejects a dark soliton on the other end of the chain that, as it returns from its excursion up the trap, hits the central chain repeating the process. This periodic evolution is an analog of the classical Newton's cradle.

New Rational Solutions for Relativistic Discrete Toda Lattice System

In the present work, we examine the soliton excitations in the relativistic Toda lattice model using the rotational expansion method, where the coupling between the lattice sites is varied. For specific choices of the coupling strength we proceed to analyze the nonlinear wave excitations arising in the model which are found to be dark, singular and periodic solitary wave profiles. These solitary wave profiles are admitted to show possible modulation in its amplitude.

Soliton Oscillations for DNA Dynamics

In this paper the author presents a review of the work developed by herself and by Professor Alwyn Scott, on the theme of solitons in DNA. Following a time-line evolutions of the facts as the author has in her memory, the paper goes through the ten years, between 1985 and 1995, of scientific collaborations with Prof. Scott. It starts from the first continuous model for Microwave Absorption Spectra of DNA, and passing through the subject of thermally generated solitons in a Toda lattice model for DNA, and it ends with the two-dimensional models for DNA dynamics.

Compacton solutions and multiple compacton solutions for a continuum Toda lattice model

Abstract Some special solutions of the Toda lattice model with a transversal degree of freedom are obtained. With the aid of Mathematica and Wu elimination method, more explicit solitary wave solutions, including compacton solutions, multiple compacton solutions, peakon solutions, as well as periodic solutions are found in this paper.

Strongly nonlinear envelope soliton in a lattice model for periodic structure

Nonlinear waves in a periodic structure are investigated numerically in terms of a modified Toda lattice model incorporating external linear elastic effect. The inclusion of the external linear elasticity causes a drastic change in wave properties, i.e. the wave packets, whose wavenumbers are determined by the relative importance of the external force to the internal force, play central roles instead of the Toda soliton. Numerically observed wave packets are well described by the nonlinear Schrödinger equation for weakly nonlinear regime, but an up-and-down asymmetry develops in the envelope for strongly nonlinear regime. The head-on collisions of such strongly nonlinear wave packets show their solitonic properties and that they can be considered as strongly nonlinear envelope solitons having an up-and-down asymmetry in amplitude.

A perturbed Toda lattice model for low loss nonlinear transmission lines

In communications, radar, and digital electronics one traditionally uses discrete components that have unavoidable discontinuities at device or package interfaces. These components are typically used by designing a resonant response in the band of interest. Such resonant responses are inadequate for high data rates and, consequently, distributed devices are a topic of great current interest in high frequency electronics. The semiconductor based distributed devices recently developed have demonstrated fast responses. However, the limiting factor in these devices is the loss introduced by the semiconductor elements that introduces the nonlinearity. Future generations of high speed electronics will require distributed devices with lower loss but equivalent nonlinearity. Our earlier work on electrically tunable coplanar waveguide (CPW) devices incorporating nonlinear dielectric thin films of strontium barium titanate has shown that compact microwave devices based on nonlinear dielectric thin films are fast, broad-band, easily-tunable, and low loss (loss tangent <10 −2). These components are at lower loss than semiconductor components at the microwave frequencies of technological interest, enabling extensive use of nonlinear wave, self-focusing and pulse-shaping concepts when modeling the transmission lines. We are investigating the application of our novel materials technology and modeling to various distributed high speed signal processing problems in such low loss nonlinear transmission lines. We have modeled the transmission lines with discrete elements using a perturbed Toda lattice. Our full numerical simulations have shown that the transmission line is able to shape an input pulse into a train of stable traveling solitons. We have developed a perturbation theory to capture the soliton dynamics. We emphasize that how to couple a signal into the transmission line is an important issue in modeling soliton generation and transmission along the line in a realistic experimental setting.

Toda Lattice Models with Boundary

We consider the soliton solutions in the one- and (1+1)-dimensional Toda lattice models with a boundary. We make use of the solutions already known on a full line by means of Hirota's method. We explicitly construct the solutions satisfying the boundary conditions. With these classical solutions, we can understand the Z ∞ -symmetric boundary condition introduced by two-soliton solutions in a natural way.

Thermodynamics of Toda lattice models: application to DNA

Our generalised Bethe ansatz method is used to formulate the statistical mechanics of the classical Toda lattice in terms of a set of coupled integral equations expressed in terms of appropriate action-angle variables. The phase space as coordinatised by these action-angle variables is constrained; and both the soliton number density and the soliton contribution to the free energy density can be shown to decouple from the phonon degrees of freedom and to depend only on soliton-soliton interactions. This makes it possible to evaluate the temperature dependence of the soliton number density which, to leading order, is found to be proportional to T 1 3 .

On a Toda lattice model with a transversal degree of freedom. Sufficient criterion of blow-up in the continuum limit

The existence of blow-up for the continuum limit model of the Toda lattice with a transversal degree of freedom is proved analytically. It is shown that the sufficient criterion for the collapse in this model may be formulated as the requirement of non-positively of the system Hamiltonian.

Analysis of Stress-Strain Curve of Polyethylene Films by Toda-Lattice Model

Stress-strain curves of high density- and linear low density-polyethylenes and their blend films were analyzed by Toda-lattice model which has asymmetric interaction potential. Parameters in Toda-lattice model were experimentally determined from Young’s modulus and Gruneisen parameter. Young’s modulus was estimated from the initial slope of stress-strain curves, and Gruneisen parameter from the initial slope of ultrasonic velocity-strain curves. Using the parameters, we obtained the theoretical stress-strain curves at different temperatures for all the films. For all the films, the theoretical stress-strain curve was in good agreement with the experimental one except in the yielding region. Thus, nonlinear nature of polyethylene films can be ascribed mainly to the asymmetry of the interaction potential curve.

On a Toda lattice model with a transversal degree of freedom

A transversal degree of freedom is introduced in the Toda lattice. For different orders of magnitude of the longitudinal and transversal strains, coupled and uncoupled equations for these fields are derived in the discrete case as well as the continuum limit. Travelling wave solutions of the system are obtained and compared to numerical solutions. Standing waves, obtained by separation of variables, exhibit blow-up in finite time as well as bounded behaviour.

Thermally generated solitons in a toda lattice model of DNA

Here we calculate the thermal equilibrium number of solitons in DNA as a function of absolute temperature and the number of base pairs. This calculation is effected by modeling DNA as a Toda lattice with parameters chosen to match experimentally measured properties of DNA. We find that a significant number of solitons is generated at physiological temperature (310 K).

On perturbations of the periodic Toda lattice

A class of Hamiltonian systems including perturbations of the periodic Toda lattice and homogeneous cosmological models is studied. Separatrix approximation of oscillation regimes in these systems connected with Coxeter groups is obtained. Hamiltonian systems connected with simple Lie algebras are pointed out, which generalize the system describing periodic Toda lattice and allow theL -A pair representation.