New Classes of Nonlinear Evolutionary Equations

In the papers [1] and [2] some higher analogs of the nonlinear evolutionary equation of Korteweg– de Vries are considered. Some NEEs of the higher order are represented in handbook [3]. In the present paper we have constructed new types of NEEs of higher order. For their construction the methods of factorization and exact linearization were used (see [4–8]). In Section 2 the summary of results on the method of a exact linearization is given. In Section 3 the NEEs of n-th order depending on one arbitrary function and reducible to linear evolutionary equations (LEEs) by a nonlinear substitution of dependent variable are constructed. In Section 4 the NEEs of n-th order depending on two arbitrary functions are constructed. Their solutions of traveling wave type satisfying to nonlinear ODEs, linearized by transformation dependent and independent variables. In Section 5 the NEEs of n-th order depending on two arbitrary functions also are are constructed. Their stationary solutions satisfy to the nonlinear ODEs reduced to the semilinear equations.