Huygens’ principle, Dirac operators, and rational solutions of the AKNS hierarchy. (English) Zbl 1088.37035
Summary: We prove that rational solutions of the AKNS hierarchy of the form \(q = \sigma/\tau\) and \(r = \rho/\tau\), where \((\sigma,\tau,\rho)\) are certain Schur functions, naturally yield Dirac operators of strict Huygens’ type, i.e., the support of their fundamental solutions is the surface of the light-cone. This strengthens the connection between the theory of completely integrable systems and Huygens principle by extending to the Dirac operators and the rational solutions of the AKNS hierarchy, a classical result of J. E. Lagnese and K. L. Stellmacher [J. Math. Mech. 17, 461–472 (1967; Zbl 0154.36002)] concerning perturbations of wave operators.
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35Q58 | Other completely integrable PDE (MSC2000) |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
Dirac operators; fundamental solutions; Huygens’ principle; integrable hierarchies; rational solutionsCitations:
Zbl 0154.36002References:
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