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

Abstract

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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 differential equation ∂u∂t=aum0(∂u∂x)m1(∂2u∂x2)m2(∂3u∂x3)m3, m0,m1,m2,m3∈ℕ0, ∑j=03mj>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.

Keywords

• copolynomial
• δ-function
• partial differential equation
• Cauchy problem
• Cauchy-Stieltjes transform
• multiplication of copolynomials