Abstract
Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several 2times 2 Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev’s theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.
Highlights
Quasilinear systems of the form ut = V (u)ux (1)have been thoroughly investigated in the literature
Proof This statement is a straightforward generalisation of the analogous fact known for diagonalisable systems: if system (2) possesses n conservation laws with functionally independent densities, it is semi-Hamiltonian, see (Sévennec 1994)
Delta-functional reduction in the kinetic equation for soliton gas was first obtained in Pavlov et al (2012), to make the paper more self-contained, below we present a short derivation of system (10)
Summary
Have been thoroughly investigated in the literature. Here u = (u1, . . . , un)T is a column vector of the dependent variables and V is a n × n matrix. Under the additional condition that the Haantjes tensor of matrix V vanishes, any such system can be reduced to a diagonal form, rti = vi (r )rxi ,. Introducing the notation ai j = v vj −ri jvi and rewriting the equations for commuting flows in the form wri j = ai j (w j − wi ), (4) the requirement of their compatibility, (wri j )rk = (wri k )r j , implies the integrability conditions ai j,rk = ai j a jk + aik ak j − ai j aik (5). Under conditions (5), system (4) for commuting flows possesses infinitely many solutions parametrised by n arbitrary functions of one variable (Tsarev 1985). 2, we derive Jordan block analogues of equations for commuting flows (4) and integrability conditions (5). Note that the group preserving the class of diagonal systems (2) is more narrow, generated by transformations of the form ri → Ri (ri ), functions of one variable only. This observation allows one to develop the integrability theory of such systems in full analogy with Tsarev’s theory of diagonalisable systems (2) (Tsarev 1985, 1991), by requiring the existence of a hierarchy of commuting flows
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
