Hodograph-type transformations Original Research Article

We consider second-order evolution equations of the form ut=F1(u,ux)uxx+F2(u,ux) and third order of the form ut=F1(u,ux)uxxx+F2(u,ux,uxx). For each of these equations we construct point transformations which relate them with an equation of the same form. These mappings have the property that one of the old independent variable depends on the new dependent variable. This property leads us to hodograph-type transformations. In most of the results presented both the hodograph-type transformation and the relating equations contain arbitrary functions and/or arbitrary parameters. Certain choices of these functions and parameters make one of the two relating equations linear or integrable or a well-known equation. For the second-order evolution equations we construct hodograph-type transformations which connect them with a class of linear equations which are not of the same form.