Patterns of deformations of Peregrine breather of order 3 and 4 solutions to the NLS equation with multi parameters

Abstract : In this article, one gives a classification of the solutions to the one dimensional nonlinear focusing Schrödinger equation (NLS) by considering the modulus of the solutions in the $(x, t)$ plan in the cases of orders 3 and 4. For this, we use a representation of solutions to NLS equation as a quotient of two determinants by an exponential depending on $t$. This formulation gives in the case of the order 3 and 4, solutions with, respectively 4 and 6 parameters. With this method, beside Peregrine breathers, we construct all characteristic patterns for the modulus of solutions, like triangular configurations, ring and others.
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Pierre Gaillard, Mickaël Gastineau. Patterns of deformations of Peregrine breather of order 3 and 4 solutions to the NLS equation with multi parameters. Journal of Theoretical and Applied Physics, Springer, 2016, 10 (2), pp.83-89. ⟨10.1007/s40094-015-0204-6⟩. ⟨hal-01321436⟩

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