Different Representations of the Solutions to the Cylindrical Nonlinear Schrödinger Equation
DOI:
https://doi.org/10.5890/DNC.2023.09.005Abstract
Quasi-rational solutions to the cylindrical nonlinear Schrödinger 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] 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] Hasegawa, A. (1990), Optical solitons in fibers, Tracts in Modern Physics, 116, 2nd Ed., Springer.
[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] Ablowitz, M.J. and Segur, H. (1979), On the evolution of packets of water waves, Journal of Fluid Mechanics, 92(4), 691-715.
[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] 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] 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] Gaillard, P. (2012), Wronskian representation of solutions of the NLS equation and higher Peregrine breathers, Scientific Advances, 13(2), 71-153.
[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] 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] 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] Gaillard, P. (2016), Towards a classification of the quasi rational solutions to the NLS equation, Theoretical and Mathematical Physics, 189, 1440-1449.
Article Metrics
Usage tracking begins September 1, 2026.