Generalized Solutions of First-Order Nonlinear PDEs
Most of the equations of mathematical physics, and in particular nonlinear first-order partial differential equations, are a result of idealizing and simplifying assumptions. This approach promotes the effectiveness and elegance of mathematical models that adequately reflect some important qualitative features of the physical phenomena under consideration. However, sooner or later, one has to pay the price for the simplifying assumptions. The influence of factors not taken into account sometimes is gradual, and does not affect the qualitative picture of the physical phenomenon, but sometimes it is abrupt, and the simplified model is unable to describe the real course of events.