Liouville Integrability of Zero Curvature Equations

Abstract

Let S = S(∞, −∞) be the Schwartz space, and

$${u_t} = K(u)$$

(1)

be a nonlinear evolution equation (NLEE), where u = (u _l,..., u _p ) ∈ S ^p. There are different definitions on integrability of the equation (1), we shall adopt the following two definitions:

• (A.) We call the equation (1) Lax integrable if it can be written as a zero curvature equation

  $${U_t} - {V_x} + [U,V] = 0,$$

  where U=U(u), V=V(u) are two matrices which contain u as the ‘potential’, and [U,V]=UV−VU.

• (B.) We call the equation (1) Liouville integrable if (1) it can be written as a generalized Hamiltonian equation

  $$u{}_t = J\delta H/\delta u$$

  where J is a Hamiltonian operator; and (2) it possesses an infinite number of conserved densities {H _n } that are involution in pairs:

  $$\left\{ {{H_n}} \right.,\left. {{H_m}} \right\} = 0(mod D),$$

  where

  $$\left\{ {f,} \right.\left. g \right\} = (J\delta f/\delta u).(\delta g/\delta u)$$

  and f=0 (mod D) means f = (d/dx)h for some h∈ S ^p

Supported by National Natural Science Foundation through Nankai Institute of Math.