Andrus Salupere

On the KdV soliton formation and discrete spectral analysis

Abstract Based on the numerical integration (pseudospectral method) of the KdV equation governing the evolution of an initially monochromatic wave, a detailed picture of the soliton formation process is given. Systematic analysis of spectral properties of emerging KdV solitons over a wide range of dispersion parameters is presented. The number of emerging solitons is estimated from both the numerical experiment and the IST. The dynamics of spectral densities casts more light on the soliton formation process.

On the long-time behaviour of soliton ensembles

The paper is focused on the details of the emergence of Korteweg-de Vries (KdV) solitons from an initial harmonic excitation. Although the problem is a classical one, numerical simulations over a large range of dispersion parameters in the long run have demonstrated new features: the existence of soliton ensembles including also virtual (hidden) solitons and the periodic patterns of the wave-profile maxima.

The Pseudospectral Method and Discrete Spectral Analysis

One of the focal points of the research at the Centre for Nonlinear Studies is related to the numerical simulation of the emergence, propagation and interaction of solitary waves and solitons in nonlinear dispersive media. Based on the discrete Fourier transform the pseudospectral method can be used for the numerical integration of the model equations, and the Fourier transform related discrete spectral characteristics for the analysis of the numerical results. The latter approach is called discrete spectral analysis. The main advantage of the pseudospectral method compared with the finite difference method is related to the computational costs. On the other hand, the Fourier transform related spectral characteristics carry additional information about the internal structure of the waves, which can be used for the analysis of the time-space behaviour of the numerical solutions. In the present paper several practical aspects of the application of the Fourier transform based pseudospectral method for the numerical integration of equations of different types are discussed and several examples of applications of the discrete spectral analysis are introduced.

Long-time behaviour of soliton ensembles. Part I––Emergence of ensembles

Abstract This paper is focused on the emergence of the KdV solitons from an initial harmonic excitation. In the long run this process is characterized not so much by regular soliton trains but rather by soliton ensembles. It has been shown explicitly that under indicated initial conditions the width of emerging solitons are mostly larger than the distance between maxima of wave profiles. Consequently, visible are the ensembles formed by several simultaneously interacting solitons including also hidden (virtual) solitons. The conditions for emerging such ensembles are studied over the wide range of amplitude ratios for typical dispersion parameters. Based on that analysis, it is possible to cast more light to the recurrence and periodicity in the long run (see Part II).

Propagation of sech2-type solitary waves in hierarchical KdV-type systems

A hierarchical Korteweg-de Vries-type evolution equation is applied for modelling wave propagation in dilatant granular materials. The model equation is integrated numerically under sech^2-type initial conditions using the Fourier transform-based pseudospectral method. Numerical simulations are carried out over a wide range of material parameters (two dispersion parameters and one microstructure parameter) and amplitudes of the initial wave. The analysis of the time-space behaviour of solutions results in five solution types. Besides typical KdV-like solitonic structures, wave packets were detected for some domains of material parameters.

Long-time behaviour of soliton ensembles. Part II––Periodical patterns of trajectories

Abstract The emergence and interaction of KdV solitons generated by harmonic initial conditions are studied in the long run (hundreds of recurrence time tR). After the initial train of solitons has emerged, the further process is characterised by propagation of soliton ensembles and in numerical calculations the local maxima of wave profiles can be traced (see Part I for the description of ensembles and the corresponding trajectories of single solitons vs those within an ensemble). The geometrical patterns of trajectories of those maxima are analysed. In the short run arc-like patterns are formed but in the long run regular rhombus-like patterns are detected. The backbones of those rhombi are balanced trajectories which correspond to interacting solitons whose phase-shifts to the right are balanced by phase-shifts to the left. Consequently, over long time and space intervals these trajectories can be approximated by straight lines. A rhombus in the x–t plane has the space periodicity 2π dictated by the initial excitation and time periodicity 2tP or tP, dictated by balanced trajectories and related to the recurrence time tR. Such a pattern is a steady feature of regularity in the soliton emergence process that usually is taken to be shadowed by the loss of the recurrence in the strict sense.

Korteweg-de Vries soliton detection from a harmonic input

Abstract The formation of solitons from harmonic inputs in a Korteweg-de Vries system is considered for a large range of values of the dispersion parameter. A method for detecting the number of emerging solitons from the soliton amplitude curves is presented.

On the problem of periodicity and hidden solitons for the KdV model

In continuum limit, the Fermi-Pasta-Ulam lattice is modeled by a Korteweg-de Vries (KdV) equation. It is shown that the long-time behavior of a KdV soliton train emerging from a harmonic excitation has a regular periodicity of right- and left-going trajectories. In a soliton train not all the solitons are visible, the solitons with smaller amplitude are hidden and their influence is seen through the changes of phase shifts of larger solitons. In the case of an external harmonic force several resonance schemes are revealed where both visible and hidden solitons have important roles. The weak, moderate, strong, and dominating fields are distinguished and the corresponding solution types presented.

On solitons in microstructured solids and granular materials

The problems under consideration are related to wave propagation in nonlinear dispersive media, characterised by higher-order nonlinear and higher-order dispersive effects. Particularly two problems -- wave propagation in dilatant granular materials and wave propagation in shape-memory alloys -- are studied. Model equations are KdV-like evolution equations in both cases. The types of solutions are determined and analysed. It has been found that waves in granular materials are composed by two concurrent ensembles of solitary waves and there exist a threshold parameter value governing the behaviour of stable waves in alloy type materials.

Original article: On the propagation of 1D solitary waves in Mindlin-type microstructured solids

The Mindlin model and hierarchical approach by Engelbrecht and Pastrone are used for modelling 1D wave propagation in microstructured solids. After introducing the free energy function, one gets from Euler-Lagrange equations a system of equations of motion. Making use of the slaving principle, a nonlinear hierarchical wave equation can be derived. Equations are solved numerically under localized initial conditions. For numerical integration, the pseudospectral method based on the Fourier transform is used. The influence of free energy parameters on the character of dispersion and wave propagation is studied. Numerical results of hierarchical approximation and the full equation system will be compared and the quality of the approximation will be discussed.