On the remarkable relations among PDEs integrable by the inverse spectral transform method, by the method of characteristics and by the Hopf-Cole transformation

TL;DR

This paper uncovers profound connections among various classes of integrable PDEs, linking their integrability properties through matrix reductions and transformations, thereby unifying different integrability methods in a comprehensive framework.

Contribution

It demonstrates that integrability features of complex PDEs can be derived from simpler, well-understood hierarchies using matrix reductions and transformations, revealing a unified structure among diverse integrable equations.

Findings

Integrability of (2+1)-D PDEs can be generated from the (1+1)-D matrix Burgers hierarchy.

Multidimensional integrable PDEs relate to a matrix equation solvable by characteristics.

Generalized matrix reductions produce PDEs with combined integrability properties.

Abstract

We establish deep and remarkable connections among partial differential equations (PDEs) integrable by different methods: the inverse spectral transform method, the method of characteristics and the Hopf-Cole transformation. More concretely, 1) we show that the integrability properties (Lax pair, infinitely-many commuting symmetries, large classes of analytic solutions) of (2+1)-dimensional PDEs integrable by the Inverse Scattering Transform method ($S$-integrable) can be generated by the integrability properties of the (1+1)-dimensional matrix B\"urgers hierarchy, integrable by the matrix Hopf-Cole transformation ($C$-integrable). 2) We show that the integrability properties i) of $S$-integrable PDEs in (1+1)-dimensions, ii) of the multidimensional generalizations of the $GL(M,\CC)$ self-dual Yang Mills equations, and iii) of the multidimensional Calogero equations can be generated by the integrability properties of a recently introduced multidimensional matrix equation solvable by the method of characteristics. To establish the above links, we consider a block Frobenius matrix reduction of the relevant matrix fields, leading to integrable chains of matrix equations for the blocks of such a Frobenius matrix, followed by a systematic elimination procedure of some of these blocks. The construction of large classes of solutions of the soliton equations from solutions of the matrix B\"urgers hierarchy turns out to be intimately related to the construction of solutions in Sato theory. 3) We finally show that suitable generalizations of the block Frobenius matrix reduction of the matrix B\"urgers hierarchy generates PDEs exhibiting integrability properties in common with both $S$- and $C$- integrable equations.