Hodograph Transform Transform And Hodograph-type Promotion

There are many nonlinear problems in natural sciences. We can study them easily when the nonlinear problems are changed to the linear ones, so the linearizing methods are very important to us. In this text, hodograph transfor-mation and hodograph-type transformation are two methods which can change the nonlinear problems to the linear ones.We study different kinds of hodograph transformation and hodograph-type transformation in this text. For a first-order partial differential equation, hodo-graph transformation interchanges the roles of independent variables and depen-dent variables to linearize the equation. In this paper we generalise the results of hodograph transformation and pay more attentions to the linearizing problems of genuinely nonlinear equations and quasi-linear equations with four indepen-dent variables and four dependent variables. Hodograph-type transformation is the generalization of hodograph transformation. For the fourth order evolution equations of the form (?) we can construct point transformations which transform them to equations of the same form. In these point transformations one of old independent variables must depend on the new dependent variables and these are hodograph-type transformations. Certain arbitrary functions and parameters are contained in the hodograph-type trans-formation and its relating equations. Choosing these functions and parameters we will get a linear or a well-known equation.There are three main parts in this paper:1. Describing briefly the history and importance of the hodograph trans-formation and hodograph-type transformation, we introduce the symbols which will use in this text. 2. Deriving the linearizable systems with four independent variables and four dependent variables.3. Applying the hodograph-type transformation to the fourth order evolu-tion equations.