Matrix equations of hydrodynamic type as lower-dimensional reductions of Self-dual type S-integrable systems

2007/08/15 by A. I. Zenchuk, Zenchuk, A. I.
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Waves and Solitons #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.0708.2050

openalex publication_date 2007/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that matrix Q× Q Self-dual type S-integrable Partial Differential Equations (PDEs) possess a family of lower-dimensional reductions represented by the matrix Q × n₀ Q quasilinear first order PDEs solved in \citeSZ1 by the method of characteristics. In turn, these PDEs admit two types of available particular solutions: (a) explicit solutions and (b) solutions described implicitly by a system of non-differential equations. The later solutions, in particular, exhibit the wave profile breaking. Only first type of solutions is available for (1+1)-dimensional nonlinear S-integrable PDEs. (1+1)-dimensional N-wave equation, (2+1)- and (3+1)-dimensional Pohlmeyer equations are represented as examples. We also represent a new version of the dressing method which supplies both classical solutions and solutions with wave profile breaking to the above S-integrable PDEs.

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