Riemann-Hilbert problem and long-time asymptotics of the Yajima-Oikawa equation

The Yajima-Oikawa equation is an integrable long wave-short wave resonance interaction model arising as a deformation of the Zakharov system for Langmuir waves coupled to ion-acoustic waves.

In this work, a Riemann-Hilbert approach is developed for the Cauchy problem for the Yajima-Oikawa equation with rapidly decaying initial data.

A main novelty is the formulation of a direct and inverse scattering theory adapted to its third-order spectral problem, including a detailed treatment of the singular spectral point \(k=0\).

The associated Riemann-Hilbert problem is expressed in terms of two reflection coefficients determined by the initial data, together with possible discrete eigenvalues and norming constants.

We prove a vanishing lemma which ensures the unique solvability of the Riemann-Hilbert problem under suitable positivity assumptions, and hence obtain a rigorous reconstruction formula for the solution.

We also classify the admissible discrete spectrum and derive exact pure soliton solutions from the reflectionless Riemann-Hilbert problem.

In the solitonless case, we apply the Deift-Zhou nonlinear steepest descent method to obtain rigorous long-time asymptotic formulas in the different regions of the upper \((x,t)\)-plane.

The leading oscillatory behavior of the short-wave component is described explicitly in terms of the reflection coefficients evaluated at the stationary phase points, while the long-wave component is shown to be of lower order away from the transition region.

These results provide, to the best of our knowledge, the first Riemann-Hilbert framework for the long-time asymptotic analysis of the Yajima-Oikawa equation in the presence of continuous spectrum.