On a Reduction of Nonlinear Partial Differential Equations of Briot-Bouquet Type

Let F(t, x, u, v) be a holomorphic function in a neighborhood of the origin of C-4 satisfying F(0, x, 0, 0) equivalent to 0 and (partial derivative F/partial derivative v)(0, x, 0, 0) equivalent to 0; then the equation (A) t partial derivative u/partial derivative t = F(t, x, u, partial derivative u/partial derivative x) is called a partial differential equation of Briot-Bouquet type with respect to t, and the function lambda(x) = (partial derivative F/partial derivative u)(0, x, 0, 0) is called the characteristic exponent. In [15], it is proved that if lambda(0) is not an element of (-infinity, 0] boolean OR {1, 2, ...} holds the equation (A) is reduced to the simple form (B-1) t partial derivative w/partial derivative t = lambda(x)w. The present paper considers the case lambda(0) = K is an element of {1, 2, ...} and proves the following result: if lambda(0) = K is an element of [1, 2, ...) holds the equation (A) is reduced to the form (B-2) t partial derivative w/partial derivative t = lambda(x)w + gamma(x)t(K) for some holomorphic function gamma(x). The reduction is done by considering the coupling of two equations (A) and (B-2), and by solving their coupling equations. The result is applied to the problem of finding all the holomorphic and singular solutions of (A).