Hodograph-type transformations (Q1413246)

Evolution equations of the forms \(u_t=F_1(u,u_x)u_{xx}+F_2(u,u_x)\) and \(u_t=F_1(u,u_x)u_{xxx}+F_2(u,u_x,u_{xx})\) are considered in this work. For each of these equations, sufficient conditions to exist Hodograph-type transformations taking the equation to an equation of the same form are investigated. For the second order equation under consideration, Hodograph-type transformations reducing the equation to a linear equation of the form \(u_t=G_1(x)u_{xx}+G_2(x)u_x\) are constructed.