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By a copolynomial we mean here a linear functional on the space of polynomials $\mathbb{R}[x]$. The space of copolynomials is denoted by $\mathbb{R}[x]'$. If $T\in\mathbb{R}[x]'$ and $p\in\mathbb{R}[x]$, then the result of application of $T$ to $p$ is written as $(T,p)$. The derivative $T'$ of a copolynomial $T\in\mathbb{R}[x]'$ is defined in the same way as in the classical theory of generalized functions: $(T',p)=-(T,p')$, $p\in\mathbb{R}[x]$. An important example of a copolynomial is the $\delta$-function, which is defined by $(\delta,p)=p(0)$, $p\in\mathbb{R}[x]$.
The Cauchy-Stieltjes transform of a copolynomial $T\in\mathbb{R}[x]'$ is defined as the following formal Laurent series in the ring $s^{-1}\mathbb{R}[[s^{-1}]]$: $C(T)(s)=\sum_{k=0}^{\infty}(T,x^k)/s^{k+1}$. The mapping $C:\mathbb{R}[x]'\to s^{-1}\mathbb{R}[[s^{-1}]]$ is an isomorphism of these vector spaces. The multiplication of copolynomials is defined through the multiplication of their Cauchy-Stieltjes transforms. The theory of linear and nonlinear PDEs in the ring $\mathbb{R}[x]'[[t]]$ was studied in [1,2], respectively.
Let $f(z)=\sum_{j=1}^{\ell}f_jz^j\in\mathbb{R}[z]$. We prove that the following Cauchy problem for the first order quasilinear equation
$$ \frac{\partial u}{\partial t}+f(u)\frac{\partial u}{\partial x}=0, \qquad u(0,x)=\delta(x), $$ in $\mathbb{R}[x]'[[t]]$ has a unique solution $$ u(t,x)=\sum_{k=0}^{\infty}\sum_{|\alpha|=k} f_1^{\alpha_1}\ldots f_{\ell}^{\alpha_{\ell}} \frac{\left(\sum_{j=1}^{\ell}j\alpha_j+k\right)!} {\alpha_1!\ldots\alpha_{\ell}! \left(\sum_{j=1}^{\ell}j\alpha_j+1\right)!} \delta^{\sum_{j=1}^{\ell}j\alpha_j+k+1}(x)t^k, $$ where $\alpha=(\alpha_1,\ldots,\alpha_{\ell})\in\mathbb{N}_0^{\ell}$ is a multi-index, $|\alpha|=\sum_{j=1}^{\ell}\alpha_j$.
Acknowledgement. Roman Skurikhin was supported by the National Research Foundation of Ukraine, project No. 2025.07/0369 “Qualitative methods of nonlinear analysis of heterogeneous structures”.
[1] Gefter S. L., Piven' A. L.: Partial differential equations in module of copolynomials over a commutative ring, J. Math. Phys. Anal. Geom. 21 (2025), No. 1, 56–83.
[2] Gefter S. L., Piven' A. L.: Nonlinear Partial Differential Equations in Module of Copolynomials over a Commutative Ring. J. Math. Phys. Anal. Geom. 21 (2025), No. 3, 319–345.
