We consider the third-order partial differential equation

${w}_{tt}={[\sigma ({w}_{x})+\lambda ({w}_{x}){w}_{tx}]}_{x},$

(1)

where *σ* and *λ* are smooth functions, $w(t,x)$ is the dependent variable and subscripts denote partial derivative with respect to the independent variables *t* and *x*.

Equation (1) can describe the behavior of a one-dimensional viscoelastic medium in which nonlinearities appear not only in the elastic part of the stress, but also in the viscoelastic one.

Some mathematical questions as the global existence, uniqueness and stability of solutions can be found in [1, 2]. Moreover, shear wave solutions are found in [3], where some explicit examples of blow-up for boundary value problems with smooth initial data are shown. A symmetry analysis and some exact solutions are shown in [4–6], while when $\lambda ({w}_{x})={\lambda}_{0}$, with ${\lambda}_{0}$ a positive constant, a symmetry analysis can be performed in [7–9].

It is well known that a small dissipation is able to prevent the breaking of the wave profile allowing to study the so called ‘far field’, and a technique widely used is the perturbation analysis performed by expanding the dependent variables in power series of a small parameter (may be a physical parameter or often artificially introduced). Having in mind to perform an ‘approximate symmetry analysis’, we introduce in (1) a small parameter

*ε*, namely

${w}_{tt}=f({w}_{x}){w}_{xx}+\epsilon {[\lambda ({w}_{x}){w}_{tx}]}_{x},$

(2)

with

$f={\sigma}^{\prime}$ (hereafter, a prime denotes derivative of a function with respect to the only variable upon which it depends). For

$\epsilon =0$, we recover the nonlinear wave equation

${w}_{tt}=f({w}_{x}){w}_{xx}.$

(3)

The combination of the Lie group theory and the perturbation analysis gives rise to the so-called approximate symmetry theories. The first paper on this subject is due to Baikov, Gazizov and Ibragimov [10]. Successively another method for finding approximate symmetries was proposed by Fushchich and Shtelen [11]. In the method proposed by Baikov, Gazizov and Ibragimov, the Lie operator is expanded in a perturbation series so that an approximate operator can be found. But the approximate operator does not reflect well an approximation in the perturbation sense; in fact, even if one uses a first-order approximate operator, the corresponding approximate solution could contain higher-order terms.

In the method proposed by Fushchich and Shtelen the dependent variables are expanded in a perturbation series; equations are separated at each order of approximation and the approximate symmetries of the original equations are defined to be the exact symmetries of the system coming out from equating to zero the coefficients of the smallness parameter. This method is consistent with the perturbation theory and yields correct terms for the approximate solutions but a ‘drawback’ is present: it is impossible to work in hierarchy, *i.e.*, in the search for symmetries, there is a coupled system between the equations at several orders of approximation, therefore the algebra can increase enormously.

In this paper we work in the framework of the approximate method proposed in [12, 13], in which the expansions of the dependent variable are introduced also in the Lie group transformations so that one obtains an approximate Lie operator which permits to solve in hierarchy the invariance conditions starting from the classification of unperturbed equation (3). We obtain the symmetry classification of the functions $f({w}_{x})$ and $\lambda ({w}_{x})$ through which equation (2) is approximately invariant and search for approximate solutions.

The plan of the paper is the following. The approximate symmetry method is introduced in the next section; the group classification via approximate symmetries is performed in Section 3; in Section 4, in a physical application, the approximate solution is calculated by means of the approximate generator of the first-order approximate group of transformations.