[Paper Review] Global solution of the Wadati-Konno-Ichikawa equation with small initial data

Yusuke Shimabukuro|arXiv (Cornell University)|Dec 22, 2016

Nonlinear Waves and Solitons17 references3 citations

TL;DR

This paper establishes the global existence of solutions to the Wadati-Konno-Ichikawa (WKI) equation for small initial data in smooth weighted Sobolev spaces using inverse scattering theory. By formulating a matrix Riemann-Hilbert problem in the context of the WKI spectral problem, the authors prove long-time existence and derive exact one-soliton and bursting soliton solutions, confirming the integrability and soliton dynamics of the equation under small data assumptions.

ABSTRACT

We show the existence of global solution for the Wadati-Konno-Ichikawa (WKI) equation with small initial data in the smooth function space. Our approach is based on the scattering and inverse scattering maps in the weighted Sobolev spaces for the WKI spectral problem. In addition, we derive one soliton solution as well as a bursting soliton from our matrix Riemann-Hilbert problem.

Motivation & Objective

• To establish the global existence of solutions to the Wadati-Konno-Ichikawa (WKI) equation for small initial data in smooth function spaces.
• To analyze the scattering and inverse scattering maps in weighted Sobolev spaces to ensure solvability of the inverse problem.
• To derive exact one-soliton and bursting soliton solutions using a matrix Riemann-Hilbert problem formulation.
• To clarify the conditions under which blow-up or singularity formation occurs, particularly for the bursting soliton with infinite amplitude.

Proposed method

• Utilizes the inverse scattering transform based on the WKI spectral problem with a Lax pair formulation involving complex parameter λ and matrix potential M.
• Applies the Deift-Zhou nonlinear steepest descent method to analyze the matrix Riemann-Hilbert problem in the spectral parameter plane.
• Imposes smoothness and decay conditions on initial data via weighted Sobolev spaces Xₙ and X∞ to ensure regularity of scattering data.
• Constructs the solution via the Riemann-Hilbert problem by solving for jump matrices and asymptotic behavior at infinity and poles.
• Derives soliton solutions by analyzing poles in the spectral parameter and computing residues in the Riemann-Hilbert problem.
• Reconstructs the potential q from the solution of the Riemann-Hilbert problem using the limit of z(m(z) - I) as z → ∞.

Experimental results

Research questions

• RQ1 Under what conditions on initial data does the WKI equation admit a global solution?
• RQ2 Can the inverse scattering method be applied to the WKI equation in weighted Sobolev spaces to ensure global existence?
• RQ3 What is the structure of one-soliton and bursting soliton solutions in the WKI framework?
• RQ4 How does the Riemann-Hilbert problem formulation simplify the derivation of exact soliton solutions compared to earlier methods?
• RQ5 What role does the spectral parameter's pole structure play in generating singular soliton behavior, such as bursting solitons?

Key findings

• Global solutions exist for the WKI equation with small initial data in the smooth function space X∞, established via inverse scattering in weighted Sobolev spaces.
• The one-soliton solution is derived explicitly from the matrix Riemann-Hilbert problem, matching the form found in earlier work but with a simpler derivation.
• A bursting soliton solution is obtained when |ξ| = |η| > 0, characterized by infinite amplitude at a point in space, confirming its singular nature.
• The solution exhibits a self-consistent shift ε(x,t) defined implicitly by a nonlinear equation involving hyperbolic functions and parameters ξ, η.
• The magnitude |q(x,t)| reaches infinity in the bursting soliton case, while remaining bounded for |ξ| > |η|, confirming the transition between smooth and singular solitons.
• The method successfully reconstructs the potential q from the Riemann-Hilbert problem, with the (1,2) component of the normalized solution giving rise to the soliton profile.