Analysis of complete integrability of a system inverse to the nonlinear Benney-Kaup system (Q1332065)

The proof of complete integrability is given for the system of PDE's which is obtained by transposition of the variables \(x\) and \(t\) in the other known completely integrable system, viz., the Benney-Kaup system: \[ u_ t = u_{xx} + v_ x - uu_ x, \quad v_ t = - v_{xx} + (uv)_ x. \] The operators which constitute the Lax pair for this inverse Benney-Kaup system are constructed explicitly, which is the basis of the exact integrability of the system. Also, an infinite set of functionally independent conservation laws in involution is found. A bi-Hamiltonian representation of the system, which is another generic property of the completely integrable systems, is explicitly constructed, too.