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On zeros of the scattering data of the initial value problem for the short pulse equation. (English) Zbl 1417.35082

Summary: The short pulse equation is an important integrable equation, which can be solved by the so-called inverse scattering method with the help of a Wadati-Konno-Ichikawa type Lax pair [M. Wadati et al., J. Phys. Soc. Japan 46, No. 6, 1965–1966 (1979; Zbl 1334.81106)], in nonlinear optical field. In a recent paper [the author, J. Differ. Equations 265, No. 8, 3494–3532 (2018; Zbl 1394.35308)], we obtain the leading order long-time asymptotics of the initial value problem for the short pulse equation under the assumption that the scattering data \(a(k)\) has no zero on the above half-plane for the complex variable \(k\). In this paper, we show two different results for the zeros of the scattering data \(a(k)\) with the finite compact support initial value. Firstly, in the linear initial value case, \(a(k)\) has no zeros. Secondly, in the even box-type linear initial value, \(a(k)\) has infinite simple zeros with the same image part on the upper-half plane.

MSC:

35Q15 Riemann-Hilbert problems in context of PDEs
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35B40 Asymptotic behavior of solutions to PDEs
78A60 Lasers, masers, optical bistability, nonlinear optics
81V80 Quantum optics

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