characteristics don't intersect. If the characteristics do intersect, then a classical solution to the PDE does not exist and leads to the formation of a shock wave. Whether characteristics can intersect or not depends on the initial condition. In fact, the breaking time before a shock wave can be formed is given by Inviscid Burgers' equation for linear initial condition Subrahmanyan Chandrasekhar provided the explicit solution in 1943 when the initial condition is linear, i.e., , where a and b are constants. The explicit solution is This solution is also the complete integral of the inviscid Burgers' equation because it contains as many arbitrary constants as the number of independent variables appearing in the equation. Using this complete integral, Chandrasekhar obtained the general solution described for arbitrary initial conditions from the envelope of the complete integral. Viscous Burgers' equation The viscous Burgers' equation can be converted to a linear equation by the Cole–Hopf transformation, which turns it into the equation which can be integrated with respect to to obtain where is a function that depends on boundary conditions. If identically (e.g. if the problem is to be solved on a periodic domain), then we get the diffusion equation The diffusion equation can be solved, and the Cole–Hopf transformation inverted, to obtain the solution to the Burgers' equation: Other forms Generalized Burgers' equation The generalized Burgers' equation extends the quasilinear convective to more generalized form, i.e., where is any arbitrary function of u. The inviscid equation is still a quasilinear hyperbolic equation for and its solution can be constructed using method of characteristics as before. Stochastic Burgers' equation Added space-time noise , where is an Wiener process, forms a stochastic Burgers' equation This stochastic PDE is the one-dimensional version of Kardar–Parisi–Zhang equation in a field upon substituting . See also Euler–Tricomi equation Chaplygin's equation Conservation equation Fokker–Planck equation References External links Burgers' Equation at EqWorld: The World of Mathematical Equations. Burgers' Equation at NEQwiki, the nonlinear equations encyclopedia. Conservation equations Equations of fluid dynamics Fluid dynamics