Abstract
In the present contribution, the Hirota-Satsuma coupled KdV hierarchy on noncommutative phase-space is investigated using the noncommutative extension of Lax pair generating technique. In particular, the explicit representation of its associated Lax pair operators is constructed. It is shown that the obtained results for phase-space noncommutativity case reduce back to the standard commutative case when θ goes to ½.
Highlights
The following system of equations (u)= 2 uxxx + 3uux − 6vvx, v t− vxxx − 3uvx (1.1)is known as the Hirota–Satsuma coupled KdV system (HS-coupled KdV) proposed by Hirota and Satsuma [1], which describes interactions of two long waves with different dispersion relations
In commutative case, it is well known that the existence of Lax pair operators (L, T ) for nonlinear differential equations is a strong indication of integrability, that is, the existence of a complete set of conservation laws, of multisoliton solutions and so on
We have presented a systematic study of HS-coupled KdV hierarchy on noncommutative phase-space by using the noncommutative Lax pair generating technique
Summary
Is known as the Hirota–Satsuma coupled KdV system (HS-coupled KdV) proposed by Hirota and Satsuma [1], which describes interactions of two long waves with different dispersion relations. We make a short review of the noncommutative Lax pair generating technique [26,27,28] which is relevant for this work. This technique will be useful to derive the noncommutative extension of the HS-coupled KdV hierarchy and obtain its associated Lax pair operators. 2. Noncommutative Hirota–Satsuma coupled KdV hierarchy An NC partial differential equation which has the Lax representation can be reformulated as follows:. Let us note that the same analysis used for n = 0 and n = 1 is extended to build the 5th-order, 7th-order and the 9th-order NC HS-coupled KdV equations and their associated Lax pair operators.
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