Toda lattice mass transport in Lagrangian mechanics and in a two-dimensional system

This paper explores the connection between the hydrodynamic mass transport description and the thermodynamic description for a nonlinear range of the Toda lattices. Particular attention is paid to the broken isotropy in the KdV and Burgers equations. The flow variable representation is established from the Lagrangian mechanics for hydrodynamic mass transport. Based on the inverse scattering transform, the Gel'fand-Levitan-Marchenko (GLM) equation is formulated from the KdV equation expressed by the flow variable representation. We found that a kernel of the GLM equation is given by the concentration variable Q(x,t). A Lagrangian is formulated for the KdV equation in state space (Q(x,t),K(x,t)). Next, an extension of the flow variable representation is sought in a two-dimensional system. The LHS of the Kadomtsev-Petviashvili (KP) equation takes the same form as in the second formalism of the KdV equation. By setting up the flow variable representation of the KP equation, the Burgers equation in two dimensions is formulated. These results contribute to an understanding of the broken isotropy for the nonlinear mass transport equations. These results provide physical insight into various consequences of the generalized form of the Kawasaki-Ohta equation from the viewpoint of mass transport.