TY - JOUR

T1 - Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

AU - Al Sakkaf, Laila

AU - Al Khawaja, Usama

N1 - Funding Information:
We acknowledge the support of UAE University through grants UAEU‐UPAR(4) 2016 and UAEU‐UPAR(6) 2017.
Funding Information:
We acknowledge the support of UAE University through grants UAEU-UPAR(4) 2016 and UAEU-UPAR(6) 2017.
Publisher Copyright:
© 2020 John Wiley & Sons, Ltd.

PY - 2020/11/30

Y1 - 2020/11/30

N2 - 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.

AB - 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.

KW - Manakov system

KW - N-coupled nonlinear Schrödinger equations

KW - exact solutions

KW - superposition principle

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U2 - 10.1002/mma.6685

DO - 10.1002/mma.6685

M3 - Article

AN - SCOPUS:85087919486

VL - 43

SP - 10168

EP - 10189

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

IS - 17

ER -