Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2310.08407

A quadric ansatz method for a certain class of second order PDEs

Prim Plansangkate
View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop a procedure to implement the method of quadric ansatz to a class of second order partial differential equations (PDEs), which includes the four-dimensional Kähler-Einstein equation with symmetry and the one-sided type-D Einstein equation with nonzero scalar curvature. The procedure, which reduces the PDEs to ordinary differential equations (ODEs), involves imposing additional constraints to the form of the ansatz, depending on the exact form of the PDEs. Thus its applicability varies within the class. We identify applicable subclasses of a class of modified Toda equations, one of which includes the Kähler-Einstein equation with symmetry, and give the corresponding reduced ODEs. In addition, we obtain a quadric ansatz reduction for a family of Einstein-Weyl equations in arbitrary dimensions. Lastly, we suggest a particular form of equations for which our procedure appears to be most effective as the aforementioned constraints vanish, and also present some reduction results using another type of ansatz, namely the hyperplane ansatz.

Explore related subjects

nlin.SIgr-qcmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Prim Plansangkate. 2023-10-12. A quadric ansatz method for a certain class of second order PDEs. https://arxiv.org/abs/2310.08407

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On vector Schwarz-KdV equation

A collection of miscellaneous continuous, semi-discrete, and discrete integrable systems can be associated with each integrable evolution equation of the KdV type. We give them for the Schwarz--KdV equation and generalize to the vector case. The existence of these vector generalizations is a non-trivial experimental fact for which no mathematical explanation is yet known.

nlin.SI↗
arXiv2026

An integrable $\mathbb{Z}_2^2$-graded extension of Camassa-Holm equation and its bi-Hamiltonian structure

By constructing Lax operators in the loop algebra of the $\mathbb{Z}_2^2$-graded extension of the Lie superalgebra $\mathfrak{osp}(1|2)$, we derive a $\mathbb{Z}_2^2$-graded extension of the Camassa-Holm equation. The resulting equation is an integrable nonlinear PDE for a system of four $\mathbb{Z}_2^2$-graded commutative functions, each associated with a distinct $\mathbb{Z}_2^2$-degree. We further show that the $\mathbb{Z}_2^2$-Camassa-Holm equation admits a bi-Hamiltonian structure. As a consequence, it possesses infinitely many conserved quantities, including one with non-trivial $\mathbb{Z}_2^2$-degree, which are mutually in involution with respect to the $\mathbb{Z}_2^2$-graded Poisson brackets.

nlin.SI↗
arXiv2026

Integrability of the deformed Toda systems

In 2020 M. Mucciconi and L. Petrov introduced a long-range deformation of the quantum open non-relativistic Toda system. We prove the integrability of the deformed Toda system by constructing a $2 \times 2$ Lax operator, which produces the commutative family of differential operators containing the Hamiltonian of the deformed Toda system. Moreover, we show that the same integrable deformation exists on both classical and quantum levels and can be applied to both non-relativistic and relativistic Toda systems. For the open non-relativistic deformed Toda systems we also present an $n \times n$ Lax matrix and prove that it produces the same family of Hamiltonians. We also show how to obtain the van Diejen-type deformed Toda system. Lastly, we show that on the quantum level the algebraic Bethe ansatz technique can be applied to the deformed Toda system.

nlin.SI↗