The Crocco transformation: order reduction and construction of Bäcklund transformations and new integrable equations

Wide classes of nonlinear mathematical physics equations are described that admit order reduction through the use of the Crocco transformation, with a first-order partial derivative taken as a new independent variable and a second-order partial derivative taken as the new dependent variable. Associated Bäcklund transformations are constructed for evolution equations of general form (special cases of which are Burgers, Korteweg–de Vries, and many other nonlinear equations of mathematical physics). The results obtained are used for order reduction and constructing exact solutions of hydrodynamics equations (Navier– Stokes, Euler, and boundary layer). A number of new integrable nonlinear equations, inclusive of the generalized Calogero equation, are considered. AMS classification scheme numbers: 35Q58, 35K55, 35K40, 35Q53, 35Q30 The Crocco transformation: order reduction and new integrable equations 2 1. Preliminary remarks The Crocco transformation is used in hydrodynamics for reducing the order of the plane boundary-layer equations [1–3]. It is a transformation in which a first-order partial derivative taken as a new independent variable and a second-order partial derivative taken as the new dependent variable. So far, using the Crocco transformation has been limited solely to the theory of boundary layer. The present paper reveals that the domain of application of the Crocco transformation is much broader. It can be successfully used for reducing the order of wide classes of nonlinear equations with mixed derivatives and constructing Bäcklund transformations for evolution equations of arbitrary order and quite general form, special cases of which include Burgers and Korteweg–de Vries type equations as well as many other nonlinear equations of mathematical physics. The Bäcklund transformations obtained with the Crocco transformation may, in turn, be used for constructing new integrable nonlinear equations. Examples of the generalized Calogero equation and a number of other integrable nonlinear second-, third-, and fourthorder equations are considered. A generalization of the Crocco transformation to the case of three independent variables is given. It is noteworthy that various Bäcklund transformations and their applications to specific equations of mathematical physics can be found, for example, in [3–14]. In the present paper, the term integrable equation applies to nonlinear partial differential equations that admit solution in terms of quadratures or solutions to linear differential or linear integral equations. 2. Nonlinear equations that admit order reduction with the Crocco transformation Consider the nth-order nonlinear equation with a mixed derivative ∂u ∂t∂x + [a(t)u+ b(t)x] ∂u ∂x2 = F ( t, ∂u ∂x , ∂u ∂x2 , ∂u ∂x3 , . . . , ∂u ∂xn ) . (1) 1. General property: if ũ(t, x) is a solution to equation (1), then the function u = ũ(t, x+ φ(t)) + 1 a(t) [b(t)φ(t)− φ′t(t)], a(t) 6≡ 0, (2) where φ(t) is an arbitrary function, is also a solution to equation (1). If a(t) ≡ 0, then u = ũ(t, x) + φ(t) is another solution to (1).