Hypercomplex Method for Solving Nonlinear PDEs
Abstract
A hypercomplex method for constructing solutions of a class of nonlinear partial differential equations is developed. A family of two-dimensional commutative associative algebras is introduced in which the structural coefficients depend on a component of the monogenic function itself. A corresponding theory of monogenic functions is developed, and nonlinear analogues of the Cauchy-Riemann conditions are obtained. It is shown that the first component of a monogenic function satisfies a broad class of second-order nonlinear partial differential equations whose coefficients depend on the unknown function. A hodograph transformation is used to linearize the nonlinear Cauchy-Riemann system. As a result, the problem of constructing solutions of the nonlinear equations is reduced to solving a second-order linear partial differential equation with variable coefficients, followed by the inverse hodograph transformation. A local correspondence between solutions of the nonlinear Cauchy-Riemann system and solutions of the associated linear equation is established, yielding a constructive algorithm for generating solutions of the original nonlinear PDEs. Several examples are presented in which explicit and implicit solutions of nonlinear second-order PDEs are obtained by this method.
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BibTeX
@misc{shpakivskyi2026hypercomplex,
title = {{Hypercomplex Method for Solving Nonlinear PDEs}},
author = {Vitalii Shpakivskyi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27042},
eprint = {2610.27042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27042},
note = {CC0},
}
RIS
TY - UNPB
TI - Hypercomplex Method for Solving Nonlinear PDEs
AU - Vitalii Shpakivskyi
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27042
UR - https://arxiv.org/abs/2610.27042
AB - A hypercomplex method for constructing solutions of a class of nonlinear partial differential equations is developed. A family of two-dimensional commutative associative algebras is introduced in which the structural coefficients depend on a component of the monogenic function itself. A corresponding theory of monogenic functions is developed, and nonlinear analogues of the Cauchy-Riemann conditions are obtained. It is shown that the first component of a monogenic function satisfies a broad class of second-order nonlinear partial differential equations whose coefficients depend on the unknown function. A hodograph transformation is used to linearize the nonlinear Cauchy-Riemann system. As a result, the problem of constructing solutions of the nonlinear equations is reduced to solving a second-order linear partial differential equation with variable coefficients, followed by the inverse hodograph transformation. A local correspondence between solutions of the nonlinear Cauchy-Riemann system and solutions of the associated linear equation is established, yielding a constructive algorithm for generating solutions of the original nonlinear PDEs. Several examples are presented in which explicit and implicit solutions of nonlinear second-order PDEs are obtained by this method.
ER -