Approximate group analysis for equations of subisothermal glaciers dynamics (Q1897922)

The equation of subisothermal glaciers dynamics with small budget of ice mass is investigated. This equation is represented in the form of the following PDE system \[ u_t= (u^2 \sigma^n)_x+ \varepsilon(t, x, u, \sigma),\qquad \sigma= uu_x, \] where \(t\) is the time, \(x\) is the longitudinal coordinate, the unknown function \(u(t, x)\) describes the free surface of the glacier, \(\varepsilon\Phi\) is the budget of ice mass, and \(\varepsilon\) is a small parameter. Functions \(\Phi\) such that this system admits an approximate to within \(O(\varepsilon^2)\) one- parameter symmetry group are found. Approximate to within \(O(\varepsilon^2)\) invariant solutions of these systems are obtained.