Shielding of soliton gases for the focusing nonlocal nonlinear Schrödinger equation under nonzero background

In this paper, we investigate the shielding of soliton gas solutions for focusing nonlocal nonlinear Schrödinger equation under nonzero background. The gas of soliton is obtained from the N-soliton solution when the corresponding discrete eigenvalues fill uniformly certain domain of the complex spectral plane. By introducing the modified interpolating function for the norming constants on quadrature domains, we show that the soliton gas solution can degenerate into an effective one-soliton or multi-soliton solution in the limit N → ∞. Furthermore, we present the determinant expression for the shielding soliton solutions as the limits of N-soliton solution, associated with a finite number of equivalent eigenvalues. Specifically, we characterize the geometric configurations of quadrature domains based on the number of equivalent eigenvalues, and obtain four types of soliton shielding phenomena, including the shielding one-soliton, homoclinic one-breather, heteroclinic breather and three-soliton solutions.