Single- and double-kink solutions of a one-dimensional, viscoelastic generalization of Burgers’ equation

Abstract

PurposeThe 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/approachAn 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.FindingsIt 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/valueA 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.