[Paper Review] Defocusing Hirota equation with fully asymmetric non-zero boundary conditions: the inverse scattering transform

Rusuo Ye, Peng‐Fei Han+1 more|arXiv (Cornell University)|Jan 30, 2024

Nonlinear Waves and Solitons4 citations

TL;DR

This paper develops the inverse scattering transform (IST) for the defocusing Hirota equation with fully asymmetric non-zero boundary conditions (NZBCs), where the wave envelope approaches different non-zero amplitudes and phases at spatial infinities. By formulating the scattering problem on a single Riemann sheet and solving a matrix Riemann-Hilbert problem (RHP) on an open contour, the authors derive exact solutions, including solitons, under asymmetric boundary conditions, providing a framework applicable to nonlinear optics with unequal power levels.

ABSTRACT

The paper aims to apply the inverse scattering transform to the defocusing Hirota equation with fully asymmetric non-zero boundary conditions (NZBCs), addressing scenarios in which the solution's limiting values at spatial infinities exhibit distinct non-zero moduli. In comparison to the symmetric case, we explore the characteristic branched nature of the relevant scattering problem explicitly, instead of introducing Riemann surfaces. For the direct problem, we formulate the Jost solutions and scattering data on a single sheet of the scattering variables. We then derive their analyticity behavior, symmetry properties, and the distribution of discrete spectrum. Additionally, we study the behavior of the eigenfunctions and scattering data at the branch points. Finally, the solutions to the defocusing Hirota equation with asymmetric NZBCs are presented through the related Riemann-Hilbert problem on an open contour. Our results can be applicable to the study of asymmetric conditions in nonlinear optics.

Motivation & Objective

• To extend the inverse scattering transform to the defocusing Hirota equation under fully asymmetric non-zero boundary conditions, where the asymptotic values at ±∞ have distinct moduli and phases.
• To address the branched structure of the scattering problem explicitly without relying on Riemann surfaces, enabling a single-sheet formulation of Jost solutions and scattering data.
• To characterize the analyticity, symmetry, and discrete spectrum of the scattering data under asymmetric NZBCs.
• To derive the time evolution of scattering data and construct the Riemann-Hilbert problem for the inverse scattering.
• To recover the potential and obtain exact solutions, including soliton solutions, via the RHP formulation on an open contour.

Proposed method

• Formulate the direct scattering problem by deriving Jost solutions and scattering data on a single Riemann sheet, avoiding Riemann surface constructions.
• Analyze the analyticity and symmetry properties of eigenfunctions and scattering data, including behavior at branch points.
• Establish the time evolution of reflection coefficients, norming constants, and discrete spectrum under the Hirota equation dynamics.
• Construct a matrix Riemann-Hilbert problem (RHP) on an open contour Ξ₊ to solve the inverse scattering, with jump matrix involving scattering data and phase factors.
• Transform the RHP into an equivalent problem using a gauge transformation via matrix $X_+$, leading to a new RHP for $w = m X_+^{-1}$ with explicit residue and integral conditions.
• Recover the potential $p^*$ from the large-$z$ asymptotics of the solution matrix $m$, yielding the final solution formula (4.44) involving discrete poles and a contour integral.

Experimental results

Research questions

• RQ1 How can the inverse scattering transform be formulated for the defocusing Hirota equation when the boundary conditions at $x \to \pm\infty$ are fully asymmetric, with $|p_+| \neq |p_-|$ and $\arg p_+ \neq \arg p_-$?
• RQ2 What is the analytic structure of the Jost solutions and scattering data under such asymmetric boundary conditions, particularly regarding branch points and spectral distribution?
• RQ3 How do the scattering data and eigenfunctions evolve in time under the Hirota equation, and how can this evolution be incorporated into the Riemann-Hilbert framework?
• RQ4 Can the inverse problem be solved via a Riemann-Hilbert problem on an open contour without using Riemann surfaces, and what is the resulting solution formula?
• RQ5 What is the explicit form of the solution to the defocusing Hirota equation under fully asymmetric NZBCs, including contributions from discrete eigenvalues and continuous spectrum?

Key findings

• The scattering problem for the defocusing Hirota equation with fully asymmetric NZBCs is formulated on a single Riemann sheet, avoiding the need for multi-sheeted Riemann surfaces.
• The Jost solutions and scattering data are shown to be analytic in appropriate regions, with symmetry properties and discrete spectrum distributed according to the asymmetric boundary conditions.
• The time evolution of the scattering data is derived, showing that reflection coefficients and norming constants evolve with phase factors dependent on the boundary parameters $\mu_\pm$ and $\gamma_\pm$.
• A matrix Riemann-Hilbert problem is constructed on an open contour $\Xi_+$, with jump matrix involving $J_0$ and the transformation $X_+$, enabling the solution of the inverse problem.
• The solution of the Hirota equation is recovered via the large-$z$ asymptotics of the RHP solution, yielding an explicit formula (4.44) that includes contributions from discrete poles and a contour integral over $\Xi_+ \cup \Xi_\circ$.
• The final solution formula (4.44) explicitly incorporates soliton contributions through residues at discrete eigenvalues $z_l$, with amplitudes modulated by $c_l \exp[i(\lambda_{-,l} + \lambda_{+,l})x]$, and continuous spectrum via an integral term.