Long-time Asymptotic for the Derivative Nonlinear Schrödinger Equation with Step-like Initial Value

Abstract

Abstract. We present a new Riemann-Hilbert problem formalism for the initial value problem for the derivative nonlinear Schrödinger (DNLS) equation:

Long-time behavior of solutions to the derivative nonlinear Schrödinger equation for soliton-free initial data

“…Kitaev and Vartanian [17] as well as more recently Xu and Fan [18], considered the same problem for Schwartz class initial data in the soliton-free sector and obtain in the asymptotic formula (1.10) an error term of order (log t)/t . Our results apply to a larger class of initial data and, thanks to the ∂-approach, arguably entail a simpler proof than earlier studies of the problem.…”

Inverse Scattering in 60 Minutes

“…Kitaev and Vartanian [18] obtained these asymptotic formulae assuming "small data" in the Schwarz class. Xu and Fan [29] also obtained asymptotic formulae with similar assumptions. Although we do make spectral assumptions on the initial data, we do not need a small data condition; the examples discussed in Remark 1.6 show that there exist functions q with large L 2 norm which satisfy our spectral assumptions.…”

Long-time asymptotics for the derivative nonlinear Schrödinger equation on the half-line

“…This change of spectral parameter appeared already in [20] and was employed in [31] for the analysis of the IVP for (1.1). One advantage of working with λ is that we only have to analyze a single critical point of Φ(ζ, λ) in order to find the long-time asymptotics.…”

“…Global existence results can also be obtained via inverse scattering techniques [29]. The longtime asymptotics of the solution of the IVP can be determined by performing a nonlinear steepest descent analysis of the associated Riemann-Hilbert (RH) problem [22,31,32].…”

“…Compared with the analysis of the IVP (see [22,31]), the nonlinear steepest descent analysis of the half-line problem for (1.1) presents some additional challenges. Two of these challenges are: (a) Whereas the RH problem relevant for the IVP for (1.1) only has a jump across R ∪ iR, the RH problem relevant for the IBVP also has jumps across the diagonal lines e πi 4 R and e 3πi 4 R. The jumps across these diagonal lines can be expressed in terms of a spectral function h(λ) whose definition involves the initial and boundary values.…”