[Paper Review] Green's function of heat operator with pure soliton potential

M. Boiti, F. Pempinelli+1 more|arXiv (Cornell University)|Dec 30, 2011

Nonlinear Waves and Solitons4 references4 citations

TL;DR

This paper derives the Green's function and extended resolvent for the heat operator with a pure soliton potential in the Kadomtsev–Petviashvili II (KPII) equation framework. Using spectral theory and determinantal formulas involving N×N soliton parameters, it establishes boundedness and singularity properties in the complex spectral parameter k, with discontinuities at k = iκₙ, enabling foundational tools for inverse scattering in (2+1)-dimensional integrable systems.

ABSTRACT

The heat operator with a pure soliton potential is considered and its Green's function, depending on a complex spectral parameter k, is derived. Its boundedness properties in all variables and its singularities in the spectral parameter k are studied. A generalization of the Green's function, the extended resolvent, is also given.

Motivation & Objective

• To construct the Green's function for the heat operator with a pure soliton potential in the KPII equation framework.
• To analyze boundedness and singularity properties of the Green's function in the complex spectral parameter k.
• To generalize the Green's function into an extended resolvent for the heat operator, applicable to multisoliton solutions.
• To provide a foundation for inverse scattering theory in (2+1)-dimensional integrable systems with soliton potentials.

Proposed method

• Derives the Green's function G(x,x',k) via the relation G = e^{i k(x₁−x₁') + k²(x₂−x₂')} G̃(x,x',k), ensuring boundedness in x, x' ∈ ℝ² and k ∈ ℂ.
• Constructs the Jost and dual Jost solutions using τ-functions defined via determinants of incomplete Vandermonde matrices and diagonal exponentials.
• Applies the Binet–Cauchy formula to express τ(x), χ(x,k), and ξ(x,k) as sums over permutations of Nₐ + N_b parameters κₙ.
• Introduces the extended resolvent M(x,x';q) as a sum of continuous and discrete components M_c and M_d, with kernel defined via integrals and sum over index sets.
• Uses the reduction G(x,x',k) = e^{i k_ℜ[x₁−x₁' + 2k_ℑ(x₂−x₂')]} M(x,x'; k_ℑ, k_ℑ² − k_ℜ²) to relate the resolvent to the Green's function.
• Analyzes the domain of analyticity and discontinuities of the Green's function, identifying singularities at k = iκₙ due to the structure of the soliton potential.

Experimental results

Research questions

• RQ1 How can the Green's function of the heat operator with a pure soliton potential be explicitly constructed in the KPII equation setting?
• RQ2 What are the boundedness and analyticity properties of the Green's function in the complex spectral parameter k?
• RQ3 Where are the singularities of the Green's function located, and how do they relate to the soliton parameters κₙ?
• RQ4 How can the extended resolvent of the heat operator be defined for general N-soliton potentials with Nₐ ≠ N_b?
• RQ5 What is the relationship between the extended resolvent and the Green's function in the context of inverse scattering for KPII?

Key findings

• The Green's function G(x,x',k) is bounded in x, x' ∈ ℝ² and k ∈ ℂ, with finite limits at spatial infinity.
• The Green's function exhibits discontinuities precisely at k = iκₙ for n = 1,…,N, corresponding to the poles of the Jost solutions.
• The extended resolvent M(x,x';q) is constructed as a sum of continuous and discrete components M_c and M_d, ensuring boundedness for all real q ∈ ℝ².
• The discrete part M_d of the resolvent kernel is non-zero only when the spectral parameter q lies in regions defined by the soliton parameters κₙ and involves Heaviside functions θ(q₁ − κₘₙ) and θ(qₘₙₙ(x₂−x₂')).
• The Green's function is recovered from the extended resolvent via the reduction formula involving q = (k_ℑ, k_ℑ² − k_ℜ²), valid outside the parabola q₂ = q₁².
• The extended resolvent is a left/right inverse of the heat operator L(q) only when q avoids specific polygons in the q-plane, which are determined by the soliton parameters and Nₐ, N_b.