Artur Sergyeyev: Difference between revisions
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The search for partial differential systems in four independent variables ('''(3+1)D''' | |||
for short) that are integrable in the sense of soliton theory is a longstanding open | |||
problem of mathematical physics as according to general relativity our spacetime is | |||
four-dimensional, and thus the (3+1)D case is particularly relevant for applications. | |||
This problem is addressed in my recent article | |||
A. Sergyeyev, [https://arxiv.org/abs/1401.2122 ''New integrable (3+1)-dimensional systems and contact geometry''], | A. Sergyeyev, [https://arxiv.org/abs/1401.2122 ''New integrable (3+1)-dimensional systems and contact geometry''], | ||
Lett. Math. Phys. '''108''' (2018), no. 2, 359-376 ([https://arxiv.org/abs/1401.2122 arXiv:1401.2122]) | Lett. Math. Phys. '''108''' (2018), no. 2, 359-376 ([https://arxiv.org/abs/1401.2122 arXiv:1401.2122]) | ||
where it is hown that integrable (3+1)D 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. | |||
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 | Explicit form of two infinite families of integrable (3+1)D systems from this class with | ||
Revision as of 04:20, 4 January 2019
The search for partial differential systems in four independent variables ((3+1)D for short) that are integrable in the sense of soliton theory is a longstanding open problem of mathematical physics as according to general relativity our spacetime is four-dimensional, and thus the (3+1)D case is particularly relevant for applications.
This problem is addressed in my recent article
A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108 (2018), no. 2, 359-376 (arXiv:1401.2122)
where it is hown that integrable (3+1)D 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 (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.
You may wish to look at these slides for additional background and motivation before proceeding to the article itself.