Large-time asymptotics and IST for KdV with c x^{-1} sin(2kx) initial data

Establish the long-time asymptotic behavior of solutions to the Korteweg–de Vries equation q_t − 6 q q_x + q_{xxx} = 0 with smooth Wigner–von Neumann–type initial data q(x) ∼ c x^{-1} sin(2 k x) as x → +∞ for real constants c and k, and solve the associated inverse scattering problem for the Schrödinger operator L_q = −d^2/dx^2 + q(x), including cases where the number of negative eigenvalues of L_q may be finite, infinite, or zero.

Background

The authors motivate their paper by highlighting Matveev’s proposal concerning Wigner–von Neumann–type initial profiles that decay like c/x sin(2kx). Such data can give rise to significant complications for inverse scattering, including potentially infinite discrete spectra, and the standard IST machinery is not fully developed for this long-range class.

While the present paper develops a case paper demonstrating a new resonance asymptotic regime under certain restrictions, the general problem of obtaining long-time asymptotics and solving the IST for smooth c x^{-1} sin(2kx) initial data remains unresolved, as originally emphasized by Matveev.