Purpose

The purpose of this paper is to determine both analytically and numerically the kink solutions to a new one-dimensional, viscoelastic generalization of Burgers’ equation, which includes a non-linear constitutive law, and the number of kinks as functions of the non-linearity and relaxation parameters.

Design/methodology/approach

An analytical procedure and two explicit finite difference methods based on first-order accurate approximations to the first-order derivatives are used to determine the single- and double-kink solutions.

Findings

It is shown that only two parameters characterize the solution and that the existence of a shock wave requires that the (semi-positive) relaxation parameter be less than unity and the non-linearity parameter be less than two. It is also shown that negative values of the non-linearity parameter result in kinks with a single inflection point and strain and dissipation rates with a single relative minimum and a single, relative maximum, respectively. For non-linearity parameters between one and two, it is shown that the kink has three inflection points that merge into a single one as this parameter approaches one and that the strain and dissipation rates exhibit relative maxima and minima whose magnitudes decrease and increase as the relaxation and nonlinearity coefficients, respectively, are increased. It is also shown that the viscoelastic generalization of the Burgers equation presented here is related to an ϕ⁸−scalar field.

Originality/value

A new, one-dimensional, viscoelastic generalization of Burgers’ equation, which includes a non-linear constitutive law and relaxation is proposed, and its kink solutions are determined both analytically and numerically. The equation and its solutions are connected with scalar field theories and may be used to both studies the effects of the non-linearity and relaxation and assess the accuracy of numerical methods for first-order, non-linear partial differential equations.

中文翻译：

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伯格斯方程的一维粘弹性推广的单扭结解和双扭结解

目的

本文的目的是通过分析和数值确定 Burgers 方程的新一维粘弹性推广的扭结解，该方程包括非线性本构定律，以及作为非线性函数的扭结数和松弛参数。

设计/方法/方法

基于一阶导数的一阶精确逼近的分析程序和两种显式有限差分方法用于确定单扭结解和双扭结解。

发现

结果表明，只有两个参数表征解，并且冲击波的存在要求（半正）松弛参数小于 1，非线性参数小于 2。还表明，非线性参数的负值分别导致具有单个拐点的扭结以及具有单个相对最小值和单个相对最大值的应变和耗散率。对于 1 和 2 之间的非线性参数，结果表明，当该参数接近 1 时，扭结具有三个拐点，这些拐点合并为一个拐点，并且应变和耗散率表现出相对最大值和最小值，其幅度随着松弛系数和非线性系数分别增加。ϕ ^{8 -}标量场。

原创性/价值

提出了一种新的 Burgers 方程的一维粘弹性推广，其中包括非线性本构定律和松弛，并通过解析和数值方法确定其扭结解。该方程及其解与标量场理论相关，可用于研究非线性和弛豫的影响以及评估一阶非线性偏微分方程数值方法的准确性。