2020/07/15 by L. Al Sakkaf, Laila Al Sakkaf, U. Al Khawaja +1
Physics and Astronomy · #Composite number #Exact solutions in general relativity #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Operator (biology) #Quantum Mechanics and Non-Hermitian Physics #Rotation (mathematics) #Superposition principle #nlin.SI
paper · pdf · doi:10.1002/mma.6685
published as Mathematical Methods in the Applied Sciences, 43(17), 2020, pp.10168-10189 · 23 pages, 11 figures
openalex publication_date 2020/07/15 · openalex created_date 2020/07/23 · arxiv created 2021/08/02 · arxiv updated 2021/08/03 · openalex updated_date 2026/08/05
We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two‐coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two‐dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N ‐coupled nonlinear Schrödinger equations. Specific examples for the three‐coupled nonlinear Schrödinger equation are given.