The Gardner Equation in Elastodynamics

The Gardner equation is an integrable system which includes the Korteweg–de Vries (for quadratic nonlinearitry) and the modified Korteweg–de Vries equation (for cubic nonlinearity) as special cases. It is well known that shear waves in isotropic elasticity are usually attained by introducing cubic nonlinearities in the constitutive assumptions. Here, by considering different per- turbative limits, within the classical Mooney–Rivlin energy, we obtain that for weakly dispersive materials in different perturbative limits, the resulting Boussinesq equations lead to the Gardner equation. Specifically this is attained in the three cases of prestrained, anisotropic, and when a sub- strate interaction is taken into account. This result allows us also to discuss the possible occurrence of flatons as solitary transverse waves.