[Paper Review] Solitary waves in one-dimensional pre-stressed lattice and its continual analog

Vsevolod Vladimirov, Sergii Skurativskyi|arXiv (Cornell University)|Dec 18, 2015

Advanced Mathematical Physics Problems8 references3 citations

TL;DR

This paper investigates compacton solutions—highly localized, stable traveling waves—in a continual analog of a one-dimensional pre-stressed granular chain. Using a multi-scale asymptotic expansion to derive a nonlinear PDE, the authors identify both bright and dark compacton solutions, demonstrate their stability via analytical and numerical methods, and show that these compactons preserve their shape after collisions, closely matching dynamics observed in discrete granular chains.

ABSTRACT

One of the most interesting phenomena occuring in nonlinear media models is the existence of wave patterns, such as kinks, solitons, compactons, peakons and many others. There are known numerous nonlinear evolutionary PDEs, supporting soliton (multi-soliton) and compacton traveling wave (TW) solutions. Unfortunately, the vast majority of the models, with the exception of completely integrable ones, do not enable to analyze the properties of solitary waves interaction using only qualitative methods. Therefore it is instructive, when dealing with the non-integrable PDEs, to combine the qualitative treatment with numerical simulations. In this report we are going to present the results of studying compacton solutions in the continual models for granular pre-stressed chains. The model is shown to possess a pair of compacton TW solutions which are the bright and dark compactons. First we consider the stability properties of the compacton solutions and show that both the bright and the dark compactons pass the stability test. Next we analyze the dynamics of the compactons, simulating numerically the temporal evolution of a single compacton, a well as the interection of pairs of compactons, including bright-bright, dark-dark and bright-dark pairs. To be able to simulate the evolutin of interacting compactons, we have modified the numerical scheme built by J. de Frutos, M. A. Lopez-Marcos, and J. M. Sanz-Serna. Results of simulations are compared with that of evolution of corresponding impulse in the granular pre-stressed chain.

Motivation & Objective

• To analyze the existence and stability of compacton solutions in a continual model derived from a one-dimensional pre-stressed granular chain.
• To investigate whether compactons in non-integrable PDEs can exhibit soliton-like behavior, such as shape preservation after collisions.
• To compare the dynamics of compactons in the continual model with numerical simulations of the underlying discrete granular chain.
• To evaluate the validity and accuracy of the continual approximation in capturing the behavior of localized pulses in discrete pre-stressed systems.

Proposed method

• Derive a nonlinear PDE from the discrete ODE system of a pre-stressed granular chain using a multi-scale asymptotic expansion and continuum limit.
• Apply a formal multi-scale decomposition to obtain a modified PDE that supports compacton solutions with long temporal and short spatial scales.
• Construct Hamiltonian representations and use variational principles to identify compacton solutions as extrema of Lagrangian functionals.
• Perform stability analysis using methods from [2, 7, 6], testing for perturbations that grow over time.
• Implement a modified numerical scheme based on de Frutos, López-Marcos, and Sanz-Serna for simulating compacton evolution and collisions.
• Compare numerical results from the continual PDE with direct simulations of the discrete granular chain under identical initial conditions.

Experimental results

Research questions

• RQ1 Do compacton solutions exist in the continual analog of a pre-stressed granular chain, and what are their analytical forms?
• RQ2 Are the derived compacton solutions stable under small perturbations, and do they maintain their shape during propagation?
• RQ3 How do interacting compactons—bright-bright, dark-dark, and bright-dark pairs—behave dynamically, particularly after collisions?
• RQ4 To what extent does the continual PDE model accurately reproduce the dynamics of localized pulses observed in the discrete granular chain?
• RQ5 Why do compacton solutions in the standard Nesterenko equation fail stability tests, while those in the multi-scale derived PDE remain stable?

Key findings

• The multi-scale derived PDE supports a pair of compacton solutions: bright and dark compactons, both of which pass the stability test.
• Numerical simulations show that both single compactons and interacting pairs (bright-bright, dark-dark, bright-dark) fully reestablish their shapes after collisions.
• The width of the compacton solution in physical space is Δx ≈ 4.96a for n = 3/2 (Hertzian contact), consistent with experimental and numerical results.
• The continual model accurately reproduces the dynamics of localized pulses in the discrete granular chain, as confirmed by direct comparison of numerical simulations.
• Compacton solutions in the standard Nesterenko equation are unstable, but the modified PDE from multi-scale analysis yields stable compactons.
• The numerical scheme is successfully adapted to simulate compacton interactions, enabling detailed study of their collision dynamics.