Parametric solutions for the Monge–Ampere equation and gas flow with variable entropy

The non-uniform Monge–Ampere equation is considered for an unknown function $Y=Y(x_1,x_2)$ of two independent variables. We set out an algorithm for building analytical solutions depending on a parameter $\alpha$: $x_1=\alpha x_2^2+x_2C_1+C_0$; $C_0$ and $C_1$ are constant. The cases $C_1=0$ and $C_1\ne0$ are studied. Totally, eleven exact partial solutions with arbitrary functions or arbitrary constants were constructed. We also present a gas-dynamical interpretation for one of the solutions, namely, the problem of shock wave propagation in a thermodynamically stable compressible medium with a nonclassic (sign-alternating) convexity in the equation of state. Two examples of gas flowing between movable impermeable pistons are built in the finite form. The first case deals with expansion of the thermodynamically anomalous gas (negative convexity of the state equation): the pistons move in opposite directions; the flow contains a rarefaction shock wave moving from the right piston to the left one; the gas behind the jump front is thermodynamically normal (positive convexity of the state equation); and the process lasts until the rarefaction shock wave front reaches the left piston. In the second case, we consider compression of the thermodynamically normal gas: the pistons move to meet each other, and a compression shock wave propagates in the gas; the gas behind the jump front is thermodynamically anomalous; and the process lasts till the moment the compression shock wave front reaches the left piston. The shock transitions represented are accompanied with emission/absorption of the momentum and energy in the vicinity of the strong jump line.