Geodesic
Lax Representation for a Triplet of Scalar Fields
Teoretičeskaâ i matematičeskaâ fizika, Tome 134 (2003) no. 3, pp. 401-415
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We construct a $(3\times3)$ matrix zero-curvature representation for the system of three two-dimensional relativistically invariant scalar fields. This system belongs to the class described by the Lagrangian $L=[g_{ij}(u)u^i_x u^j_t]/2 + f(u)$, where $g_{ij}$ is the metric tensor of a three-dimensional reducible Riemannian space. We previously found all systems of this class that have higher polynomial symmetries of the orders 2, 3, 4, or 5. In this paper, we find a zero-curvature representation for one of these systems. The calculation is based on the analysis of an evolutionary system $u_t=S(u)$, where $S$ is one of the higher symmetries. This approach can also be applied to other hyperbolic systems. We also find recursion relations for a sequence of conserved currents of the triplet of scalar fields under consideration.
Keywords: Lax representation, hyperbolic systems, higher symmetries, higher conservation laws. 
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D. K. Demskoi; A. G. Meshkov. Lax Representation for a Triplet of Scalar Fields. Teoretičeskaâ i matematičeskaâ fizika, Tome 134 (2003) no. 3, pp. 401-415. http://geodesic.mathdoc.fr/item/TMF_2003_134_3_a6/

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