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Representations of solutions of some nonlinear PDEs in the form of series in powers of the $\delta$-function
S.L. Gefter, A.L. Piven', V. N. Karazin Kharkiv National University
Let $K$ be an arbitrary integral domain with identity and let $K[x]$ be the ring of polynomials with coefficients in $K$. By a copolynomial we mean a $K$-linear mapping $T: K[x] \to K$. The module of copolynomials is denoted by $K[x]'$. If $T\in K[x]'$ and $p\in K[x]$, then the result of application $T\in K[x]'$ to $p\in K[x]$ is written as $(T,p)$. The derivative $T'$ of a copolynomial $T\in K[x]'$ is defined in the same way as in the classical theory of generalized functions: $(T',p)=-(T,p'),\quad p\in K[x]$. An important example of a copolynomial is the $\delta$-function which is defined by $(\delta,p)=p(0),\ p\in K[x]$.
The Cauchy-Stieltjes transform of a copolynomial $T\in K[x]'$ is defined as the following formal Laurent series from the ring $\frac{1}{s}K[[\frac{1}{s}]]$: $C(T)(s)=\sum\limits_{k=0}^\infty \frac{(T, x^k)}{s^{k+1}}$. The mapping
$C:K[x]'\to \frac{1}{s}K[[\frac{1}{s}]]$ is an isomorphism of $K$-modules. The multiplication of copolynomials is defined through the multiplication of their Cauchy-Stieltjes transforms.
The theory of linear PDEs over the module $K[x]'[[t]]$ was studied in [1,2]. We prove the following existence and uniqueness theorem for the Cauchy problem for some nonlinear PDEs.
Theorem. Let $K\supset \mathbb{Q}$, $a\in K$ and let $m_j\in\mathbb{N}_0\ (j=0,1,2,3)$.
Then the Cauchy problem
$\frac{\partial u}{\partial t}=a u^{m_0}\left(\frac{\partial u}{\partial x}\right)^{m_1}\left(\frac{\partial ^2u}{\partial x^2}\right)^{m_2}
\left(\frac{\partial ^3u}{\partial x^3}\right)^{m_3}$, $u(0,x)=\delta(x)$ has a unique solution in $K[x]'[[t]]$. This solution is of the form
$u(t,x)=\sum\limits_{k=0}^{\infty}u_k\delta^{nk+1}t^k$, where $u_k\in K$ and $n=\sum\limits_{j=0}^{3}(j+1)m_j-1$. Moreover, for every $t\in K$ this series converges in the topology of $K[x]'$.
As examples we consider a Cauchy problem for the Euler-Hopf equation $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0$, for a Hamilton-Jacobi type equation $\frac{\partial u}{\partial t}=\left(\frac{\partial u}{\partial x}\right)^2$ and for the Harry Dym equation $\frac{\partial u}{\partial t}=u^3\frac{\partial^3u}{\partial x^3}$.
This work was supported by the Akhiezer Foundation.
[1] S.L. Gefter, A.L. Piven', Linear Partial Differential Equations in Module of Formal Generalized Functions over Commutative Ring, J. Math. Sci., \textbf{257} (2021), No.5, 579--596.
[2] S.L. Gefter, A.L. Piven', Linear Partial Differential Equations in Module of Copolynomials of Several Variables over a Commutative Ring, http://arxiv.org/abs/2407.04122
