The Zakharov-Shabat spectral problem on the semi-line: Hilbert formulation and applications

Abstract

The inverse spectral transform for the Zakharov-Shabat equation on the semi-line x > 0 is reconsidered as a Hilbert problem. The boundary data induce an essential singularity as k → ∞ to one of the basic solutions. Then solving the inverse problem means solving a Hilbert problem with particular prescribed behavior. It is demonstrated that the direct and inverse problems are solved in a consistent way as soon as the spectral transform vanishes as O(1/k) at infinity in the whole upper half plane (where it may possess single poles) and is continuous and bounded on the real k-axis. The method is applied to stimulated Raman scattering and sine-Gordon (light cone) for which it is demonstrated that time evolution conserves the properties of the spectral transform.

The Ablowitz-Ladik system on the natural numbers with certain linearizable boundary conditions

“…An 'elbow scattering' transform was introduced in [18] to solve the IBVP for the KdV equation by formulating an appropriate Gel'fand-Levitan-Marchenko equation. Finally, a rigorous analysis of the problem was also presented in [19][20][21][22].…”

Numerical solutions of some stochastic hyperbolic wave equations including sine-Gordon equation

“…This is in distinction to the Euclidean coordinates which have been more commonly used in physics and engineering applications where the more familiar wave equation is as in (25), or more customarily written as Y tt − Y xx = − sin Y . Note that some authors have considered − sin Y on the right hand side of (37) (for example, Fokas, 1997;Leon and Spire, 2001;Leon, 2003;Pelloni, 2005).…”

Solution of the Dirichlet boundary value problem for the sine-Gordon equation

“…For such a simple case, the Zakharov-Shabat spectral transform has been shown to generate an infinite number of discrete eigenvalues with k = 0 as an accumulation point [4].…”

“…Indeed, simple explicit solution could only be obtained for b/a and c/a time independent which would immediately lead to θ 0 (t) = 0. Such does not occur in the Zakharov-Shabat spectral transform of sine-Gordon where piecewise constant boundary data in x = 0 produce an explicitely solvable evolution [4]. Then it has been shown that the spectral transform contains an infinite series of poles, accordingly with [3], see also [13].…”

“…The boundary value problem for (1.1) on the quadrant {x > 0, t > 0} has been solved in [3] and reconsidered in [4]. In both approaches it has been shown that a constant boundary datum θ(0, t) on an initial vacuum θ(x, 0) = 0 produces an infinite series of discrete eigenvalues with accumulation point at the origin.…”