The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equation

Abstract

Abstract. The Cauchy problem for the Dullin-Gottwald-Holm (DGH) equationwith zero boundary conditions (as |x| → ∞) is treated by the Riemann-Hilbert approach to the inverse scattering transform method. The approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used for further studying the properties of the solution, particularly, in studying its long-time behavior. Using the proposed formalism, smooth solitons as well as non-smooth cuspon solutions are presented.

Reverse-time type nonlocal Sasa–Satsuma equation and its soliton solutions

“…Shepelsky studied the Dullin-Gottwald-Holm equation with the Cauchy problem, using Riemann-Hilbert approach. Moreover, smooth and non-smooth cuspon solutions have been researched in [17]. Recently, Ma used the inverse scattering transform to study several types of reversespacetime nonlinear integrable equations and obtained various interesting results [18][19][20].…”

“…is the matrix Riemann-Hilbert problem for the nonlocal SS equation. According to the asymptotic properties in(17) and(25), we obtain the canonical normalization conditions…”