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Discontinuity, Nonlinearity, and Complexity 12(3) (2023) 539--553 | DOI:10.5890/DNC.2023.09.005

Pierre Gaillard

Department of Mathematics, Universit'e de Bourgogne-Franche Comt'e, Dijon, 9 Avenue Alain Savary, France

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Abstract

Quasi-rational solutions to the cylindrical nonlinear Schr\"odinger equation (CNLS) in terms of wronskians and Fredholm determinants of order $2N$ depending on $2N-2$ real parameters are given. We get multi-parametric families of quasi-rational solutions to the CNLS equation and we construct explicitly solutions of order $1$ to $5$.

References

1. [1]	Radhakrishnan, R. and Lakshmanan, M. (1995), Bright and dark soliton solutions to coupled nonlinear Schrodinger equations, Journal of Physics A: Mathematical and General, 28(9), 2683-2692.

2. [2]	Hasegawa, A. (1990), Optical solitons in fibers, Tracts in Modern Physics, 116, 2nd Ed., Springer.

3. [3]	Clarkson, P.A. and Hood, S. (1993), Symmetry reductions of a generalized, cylindrical nonlinear Schrodinger equation, Journal of Physics A: Mathematical and General, 26(1), 133-150.

4. [4]	Ablowitz, M.J. and Segur, H. (1979), On the evolution of packets of water waves, Journal of Fluid Mechanics, 92(4), 691-715.

5. [5]	Landman, M.J., Papanicolaou, G.C., Sulem, C., and Sulem, P.L. (1988), Rate of blowup for solutions of the nonlinear Schrodinger equation at critical dimension, Physical Review A, 38(8), 3837-3843.

6. [6]	Smirnov, A.I. and Fraiman, G.M. (1991), The interaction representation in the self-focusing theory, Physica D: Nonlinear Phenomena, 52(1), 2-15.

7. [7]	Gaillard, P. (2011), Families of quasi-rational solutions of the NLS equation and multi-rogue waves, Journal of Physics A: Mathematical and Theoretical, 44(43), 1-15.

8. [8]	Gaillard, P. (2012), Wronskian representation of solutions of the NLS equation and higher Peregrine breathers, Scientific Advances, 13(2), 71-153.

9. [9]	Gaillard, P. (2013), Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves, Journal of Mathematical Physics, 54(1), 013504-1-32.

10. [10]	Gaillar, P. (2015), Other 2N-2 parameters solutions to the NLS equation and 2N+1 highest amplitude of the modulus of the N-th order AP breather, Journal of Physics A: Mathematical and Theoretical, 48(14), 145203-1-23.

11. [11]	Gaillard, P. (2015), Multi-parametric deformations of the Peregrine breather of order N solutions to the NLS equation and multi-rogue waves, Advanced Research, 4, 346-364

12. [12]	Gaillard, P. (2016), Towards a classification of the quasi rational solutions to the NLS equation, Theoretical and Mathematical Physics, 189, 1440-1449.