at Silesian University in Opava.

My *research interests *include:

- integrable systems
- symmetries and conservation laws of partial differential systems
- Lax pairs, recursion operators, Hamiltonian and symplectic structures

The search for partial differential systems in four
independent variables (4D or (3+1)D

for short) that are integrable in the sense of soliton theory is a longstanding problem of

mathematical physics as our spacetime is four-dimensional and thus the (3+1)D case is

especially relevant for applications. The above article addresses this problem and proves

that integrable 4D systems are significantly less exceptional than it appeared before:

in addition to a handful of well-known important yet isolated examples like the (anti)

self-dual Yang--Mills equations there is a large new class of integrable (3+1)D systems

with Lax pairs of a novel kind related to contact geometry.

for short) that are integrable in the sense of soliton theory is a longstanding problem of

mathematical physics as our spacetime is four-dimensional and thus the (3+1)D case is

especially relevant for applications. The above article addresses this problem and proves

that integrable 4D systems are significantly less exceptional than it appeared before:

in addition to a handful of well-known important yet isolated examples like the (anti)

self-dual Yang--Mills equations there is a large new class of integrable (3+1)D systems

with Lax pairs of a novel kind related to contact geometry.

Explicit form of two infinite families of integrable
(3+1)D systems from this class

with polynomial and rational Lax pairs is given in the article. For example, system

with polynomial and rational Lax pairs is given in the article. For example, system

(40) is a new (and the only known to date) *integrable*
generalization from three to

four independent variables for the Khokhlov--Zabolotskaya equation, also known

as the dispersionless Kadomtsev--Petviashvili equation or the Lin--Reissner--Tsien

equation and having many applications in nonlinear acoustics and fluid dynamics.

four independent variables for the Khokhlov--Zabolotskaya equation, also known

as the dispersionless Kadomtsev--Petviashvili equation or the Lin--Reissner--Tsien

equation and having many applications in nonlinear acoustics and fluid dynamics.

You may wish to look at the recent slides (to download the PDF please use this link)

for additional background and motivation before proceeding to the article itself.

If you wish to learn more about my research, you are welcome to visit this web page.

Here are some *links *that I use a lot:

Back to the home page of Mathematical Institute.

**Last updated on **December 23, 2018, 1:45 CET