The rule for the number of fundamental Peregrine solitons involving multiple rogue wave states in the vector Chen-Lee-Liu nonlinear Schrödinger equation

This study investigates the physical distribution patterns of Peregrine solitons within multi-order rogue wave states and their potential applications in optical systems under the vector Chen-Lee-Liu nonlinear Schrödinger equation framework. Through non-recursive Darboux transformation, we systematically analyze the nonlinear dynamics of vector optical fields during second-harmonic generation, revealing an arithmetic progression in Peregrine soliton evolution across rogue wave orders. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math> th-order solutions, the fundamental Peregrine soliton count follows an arithmetic sequence with first term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mrow><mml:mi>n</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math> , last term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:mrow><mml:mi>n</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math> , and common difference <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math> , where each rogue wave state comprises fully decoupled Peregrine solitons (e.g., 1/2 for 1st-order, 2/4/6 for 2nd-order, and 6/9/12 for 3rd-order configurations). It is noteworthy that the emergence of nonet rogue wave states (nine Peregrine solitons) in third-order solutions breaks through the conventional even-mode constraint in second-order solutions, opening new avenues for investigating many-body nonlinear interactions in multi-channel photonic devices. These findings provide significant insights into the spatiotemporal localization characteristics of rogue waves in multi-component nonlinear media and their applications in optical sensing and quantum information processing.