Rogue wave patterns in the nonlocal nonlinear Schrödinger equation

This paper investigates rogue wave patterns in the nonlocal nonlinear Schrödinger (NLS) equation. Initially, employing the Kadomtsev–Petviashvili reduction method, rogue wave solutions of the nonlocal NLS equation, whose τ function is a 2×2 block matrix, are simplified. Afterward, utilizing the asymptotic analysis approach, we investigate the rogue wave patterns when two free parameters a2m1+1 and b2m2+1 are considerably large and fulfill the condition |a2m1+1|2/(2m1+1)=O(|b2m2+1|1/(2m2+1)). Our findings reveal that under these conditions, rogue wave solutions of the nonlocal NLS equation exhibit novel patterns, which consist of three regions, which are the outer region, the middle region and the inner region. In the outer and middle regions, only single rogue waves with singularities may occur, and their locations are characterized by roots of two polynomials from the Yablonskii–Vorob'ev polynomial hierarchies. In the inner region, a possible lower order rogue wave may appear, which can be singular or regular, depending on the values of m1,m2, the sizes of τ function, and certain free parameters. Finally, the numerical results indicate that the predicted outcomes are in close alignment with real rogue waves.