Higher-order reductions of the Mikhalev system

We consider the 3D Mikhalev system,

$$ u_t=w_x, \quad u_y= w_t-u w_x+w u_x, $$

which has first appeared in the context of KdV-type hierarchies. Under the reduction \(w=f(u)\), one obtains a pair of commuting first-order equations,

$$ u_t=f'u_x, \quad u_y=(f'^2-uf'+f)u_x, $$

which govern simple wave solutions of the Mikhalev system. In this paper we study higher-order reductions of the form

$$ w=f(u)+\epsilon a(u)u_x+\epsilon ^2[b_1(u)u_{xx}+b_2(u)u_x^2]+\cdots , $$

which turn the Mikhalev system into a pair of commuting higher-order equations. Here the terms at \(\epsilon ^n\) are assumed to be differential polynomials of degree n in the x-derivatives of u. We will view w as an (infinite) formal series in the deformation parameter \(\epsilon \). It turns out that for such a reduction to be non-trivial, the function f(u) must be quadratic, \(f(u)=\lambda u^2\), furthermore, the value of the parameter \(\lambda \) (which has a natural interpretation as an eigenvalue of a certain second-order operator acting on an infinite jet space), is quantised. There are only two positive allowed eigenvalues, \(\lambda =1\) and \(\lambda =3/2\), as well as infinitely many negative rational eigenvalues. Two-component reductions of the Mikhalev system are also discussed. We emphasise that the existence of higher-order reductions of this kind is a reflection of linear degeneracy of the Mikhalev system, in particular, such reductions do not exist for most of the known 3D dispersionless integrable systems such as the dispersionless KP and Toda equations.