Some class of nonlinear partial differential equations in the ring of copolynomials over a commutative ring

Abstract

We study the copolynomials, i.e., K-linear mappings from the ring of polynomials K[x] into the commutative ring K. With the help of the Cauchy–Stieltjes transform of a copolynomial, we introduce and examine a multiplication of copolynomials. We investigate the Cauchy problem related to the nonlinear partial differencial equation ∂u/∂t = aum0 • (∂u/∂x)m1 • (∂2u/∂x2)m2 • (∂3u/∂x3)m3 , m0, m1, m2, m3 ∈ N0, ∑3 (j=0) mj > 0, a ∈ K in the ring of copolynomials. To find a solution, we use the series of powers of the δ-function. As examples, we consider the Cauchy problem with the Euler-Hopf equation ∂u/∂t + u•(∂u/∂x) = 0, for a Hamilton–Jacobi type equation ∂u/∂t = (∂u/∂x)2 , and for the Harry Dym equation ∂u/∂t = u3 • (∂3u/∂x3).