‘Generalized Burgers' equation’ for nonlinear viscoelastic waves

This paper deals with nonlinear longitudinal waves in a viscoelastic medium in which the viscoelastic relaxation function has the form K(t) = const. t^-v (0<v<1). This sort of slow relaxation may be more appropriate for polymers than the often used exponential relaxation. For a far field evolution of unindirectional waves, a “generalized Burgers' equation” is obtained, which is of a form with the second derivative in the usual Burgers' equation replaced by the derivative of real order 1 + v. The steady shock solution and self-similar pulse solution to this equation are discussed. In both cases numerical solutions are presented and analytic results are obtained for the asymptotic behaviors of the solutions. It is found that both shock and pulse solutions rise exponentially, but in their tails they have slow, algebraic decay.