On long-time asymptotics to the nonlocal short pulse equation with the Schwartz-type initial data: Without solitons

Abstract

In this work, the initial value problem of the nonlocal short pulse (NSP) equation is studied with the Schwartz-type initial data. Our aim is to adequately study the long-time asymptotic behavior of the solution of the NSP equation in view of the initial value condition. Starting with the Lax pair of the NSP equation, we give some discussion about the NSP equation, including infinite number of conservation laws and finite-dimensional Hamiltonian function. The relevant spectral analysis is discussed, among which it is important to note that we discuss it separately due to the singularity of the spectrum. According to these results, a suitable Riemann–Hilbert (RH) problem which is used to express the solution of the NSP equation is established. By using the nonlinear steepest descent method, the original RH problem is transformed into a model RH problem, which is given by a parabolic cylinder function. The long-time asymptotic solution of the NSP equation is therefore given, especially the error term is quite different from the local situation.