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The Stability of Two, Three, and Four Wave Interactions of a Prototype System

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The stability of plane wave interactions of coupled nonlinear Schrödinger (CNLS) equations can be analyzed within a bisymplectic framework. This framework is a generalization of the Hamiltonian formulation. The current study considers a family of CNLS equations that are used as a prototype system for studying the combined interaction of unstable and stable component waves in optics. This popular family has a drawback when cast into a bisymplectic framework: the determinant controlling various types of fiber regime is zero. To solve this problem, it is proposed that a limit is taken from a more general CNLS family to the family in question. This method is then bench-marked against known stability results for the simple two plane wave interactions when amplitudes are equal and are found to agree. It is then applied to two wave interactions with unequal amplitudes as well as three and four wave interactions. The latter interactions for this particular system are not spectrally stable. By suggesting a slightly larger family of CNLS equations, it is illustrated that spectral stability can occur. This adapted prototype system may be of use in optics; in particular, to show that long-wave stability is possible given a judicious choice of parameter values.
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Document Type: Research Article

Affiliations: University of Surrey, Guildford

Publication date: January 1, 2008

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