semi-Hamiltonian Systems Of Hydrodynamic Type Research Articles

Algebraic-geometry approach to construction of semi-Hamiltonian systems of hydrodynamic type

In this paper, a class of semi-Hamiltonian diagonal systems of hydrodynamic type is constructed using algebraic-geometric methods. For such systems, hydrodynamic integrals and hydrodynamic symmetries are constructed from algebraic-geometric data. Besides, it is described what algebraic-geometric data distinguish in this class Hamiltonian diagonal systems with Hamiltonian structures defined by flat metrics (local Dubrovin-Novikov brackets) and metrics of constant curvature (nonlocal Mokhov-Ferapontov brackets).

Nonlinear wave interaction problems in the three-dimensional case

Three-dimensional nonlinear wave interactions have been analytically described. The procedure under interest can be applied to three-dimensional quasilinear systems of first order, whose hydrodynamic reductions are homogeneous semi-Hamiltonian hydrodynamic type systems (i.e. possess diagonal form and infinitely many conservation laws). The interaction of N waves was studied. In particular we prove that they behave like simple waves and they distort after the collision region. The amount of the distortion can be analytically computed.

Critical points, Lauricella functions and Whitham-type equations

A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear Darboux equations for conjugate nets. The distinguished role of the Euler–Poisson–Darboux equations and associated Lauricella-type functions is emphasised. In particular, it is shown that the classical g-phase Whitham equations for the KdV and NLS equations are obtained via a g-fold iterated Darboux-type transformation generated by appropriate Lauricella functions.

Integrable discretization of hodograph-type systems, hyperelliptic integrals and Whitham equations

Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In particular, we present discrete counterparts of (generalized) hodograph equations, hyperelliptic integrals and associated cycles, characteristic speeds of Whitham-type and (implicitly) the corresponding Whitham equations. By construction, the intimate relationship with integrable system theory is maintained in the discrete setting.

Algebro-Geometric Approach in the Theory of Integrable Hydrodynamic Type Systems

The algebro-geometric approach for integrability of semi-Hamiltonian hydrodynamic type systems is presented. This method is significantly simplified for so-called symmetric hydrodynamic type systems. Plenty interesting and physically motivated examples are investigated.

On integrability of 3×3 semi-Hamiltonian hydrodynamic type systems uit = vij( u) ujx which do not possess Riemann invariants

We give a complete classification of the integrable 3×3 semi-Hamiltonian hydrodynamic type systems which do not possess Riemann invariants. The “integrability” is understood as the existence of at least one higher integral. All these equations are transformable to the system of resonant 3-wave interaction by appropriate differential substitution and hence are actually integrable via inverse scattering transform. Considerations are based on a convenient coordinate-free approach to nondiagonalizable systems.