Multi-component Nonlinear Schrödinger Equations with Nonzero Boundary Conditions: Higher-Order Vector Peregrine Solitons and Asymptotic Estimates

• Published in: Journal of nonlinear science Vol. 31; no. 5
• Main Authors: Zhang, Guoqiang, Ling, Liming, Yan, Zhenya
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2021; Springer Nature B.V
• Subjects: Analysis; Asymptotic properties; Boundary conditions; Classical Mechanics; Economic Theory/Quantitative Economics/Mathematical Methods; Group theory; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Nonlinear systems; Parameters; Polynomials; Schrodinger equation; Solitary waves; Theoretical; Parity-time-reversal symmetry; Multi-component NLS equations; Lax pair; Loop group method; 35Q55; Asymptotic estimates; 37K40; 35Q51; 37K10; Nonzero boundary conditions; Darboux transform; 35Q15; Governing polynomial; Higher-order vector Peregrine solitons
• ISSN: 0938-8974; 1432-1467
• DOI: 10.1007/s00332-021-09735-z

The any multi-component nonlinear Schrödinger (alias n -NLS) equations with nonzero boundary conditions are studied. We first find the fundamental and higher-order vector Peregrine solitons (alias rational rogue waves (RWs)) for the n -NLS equations by using the loop group theory, an explicit n + 1 -multiple root of a characteristic polynomial of degree ( n + 1 ) related to the Benjamin–Feir instability, and inverse functions. Particularly, the fundamental vector rational RWs are proved to be parity-time-reversal symmetric for some parameter constraints and classified into n cases in terms of the degree of the introduced polynomial. Moreover, a systematic approach is proposed to study the asymptotic behaviors of these vector RWs such that the decompositions of RWs are related to the so-called governing polynomials F ℓ ( z ) , which pave a powerful way in the study of vector RW structures of the multi-component integrable systems. The vector RWs with maximal amplitudes can also be determined via the parameter vectors, which are interesting and useful in the study of RWs for multi-component nonlinear physical systems.