Discussed here is a regularized version of the classical Gardner equation \( u_t + u_x + uu_x + A u^2u_x - u_{xxt} \, = \, 0, \) that arises in hydrodynamics and plasma physics. This initial-value problem posed on all of \({\mathbb {R}}\) will be considered with bore-like initial data. That is, the initial wave configuration will consist of a moderately smooth function that asymptotes to zero as the spatial variable \(x \rightarrow +\infty \) , but converges to \(r > 0\) as \(x \rightarrow -\infty \) . Such initial profiles can arise in internal wave propagation, for example. In their idealized versions set on all of \({\mathbb {R}}\) , they possess an infinite amount of potential energy. This makes the analysis of the initial-value problem a slightly more subtle than the common situation where the initial profile is assumed to be localized, so being modelled by Sobolev-class initial data.
