Wronskian Representation of Solutions of NLS Equation, and Seventh Order Rogue Wave
Author(s)
Pierre Gaillard*
ABSTRACT
In this paper, we use the representation of the solutions of the focusing nonlinear Schrodinger equation we have constructed recently, in terms of wronskians; when we perform a special passage to the limit, we get quasi-rational solutions expressed as a ratio of two determinants. We have already construct breathers of orders N = 4, 5, 6 in preceding works; we give here the breather of order seven.
Cite this paper
P. Gaillard, "Wronskian Representation of Solutions of NLS Equation, and Seventh Order Rogue Wave," Journal of Modern Physics, Vol. 4 No. 2, 2013, pp. 246-266. doi: 10.4236/jmp.2013.42035.
P. Gaillard, "Wronskian Representation of Solutions of NLS Equation, and Seventh Order Rogue Wave," Journal of Modern Physics, Vol. 4 No. 2, 2013, pp. 246-266. doi: 10.4236/jmp.2013.42035.
References
[1] V. E. Zakharov, “Stability of Periodic Waves of Finite Amplitude on a Surface of a Deep Fluid,” Journal of Applied Mechanics and Technical Physics, Vol. 9, No. 2, 1968, pp. 86-94. [2] V. E. Zakharov and A. B. Shabat, “Exact Theory of Two Dimensional Self-Focusing and One Dimensinal Self-Modulation of Waves in Nonlinear Media,” Soviet Physics—JETP, Vol. 34, 1972, pp. 62-69. [3] A. R. Its and V. P. Kotlyarov, “Explicit Expressions for the Solutions of Nonlinear Schrodinger Equation,” Docklady Akademii Nauk SSSR, Vol. 965, No. 11, 1976, pp. 965-968 [4] D. Peregrine, “Water Waves, Nonlinear Schrodinger Equations and Their Solutions,” Journal of the Australian Mathematical Society, Vol. 25, 1983, pp. 16-43. [5] N. Akhmediev, V. Eleonsky and N. Kulagin, “Generation of Periodic Trains of Picosecond Pulses in an Optical Fiber: Exact Solutions,” Soviet Physics—JETP, Vol. 62, 1985, pp. 894-899. [6] N. Akhmediev, V. Eleonskii and N. Kulagin, “Exact First Order Solutions of the Nonlinear Schrodinger Equation,” Theoretical and Mathematical Physics, Vol. 72, No. 2, 1988, pp. 809-196. doi:10.1007/BF01017105 [7] N. Akhmediev, A. Ankiewicz and J. M. Soto-Crespo, “Rogue Waves and Rational Solutions of Nonlinear Schrodinger Equation,” Physical Review E, Vol. 80, 2009, Article ID: 026601. [8] D. J. Kedziora, A. Ankiewicz and N. Akhmediev, “Circular Rogue Wave Clusters,” Physical Review E, Vol. 84, 2011, Article ID: 056611. [9] P. Dubard, P. Gaillard, C. Klein and V. B. Matveev, “On Multi-Rogue Waves Solutions of the NLS Equation and Position Solutions of the KdV Equation,” European Physical Journal, Vol. 185, No. 1, 2010, pp. 247-258. doi:10.1140/epjst/e2010-01252-9 [10] P. Gaillard, “Families of Quasi-Rational Solutions of the NLS Equation and Multi-Rogue Waves,” Journal of Physics A: Mathematical and Theoretical, Vol. 44, No. 43, 2011, pp. 1-15. doi:10.1088/1751-8113/44/43/435204 [11] A. Ankiewicz, N. Akhmediev and P. A. Clarkson, “Rogue Waves, Rational Solutions, the Patterns of Their Zeros and Integral Relations,” Journal of Physics A: Mathematical and Theoretical, Vol. 43, 2010, pp. 1-9 [12] B. Guo, L. Ling and Q. P. Liu, “Nonlinear Schrodinger Equation: Generalized Darboux Transformation and Rogue Wave Solutions,” Physical Review E, Vol. 85, No. 2, 2012, Article ID: 026607. doi:10.1103/PhysRevE.85.026607 [13] Y. Ohta and J. Yang, “General High-Order Rogue Waves and Their Dynamics in the Nonlinear Schrodinger Equation,” Proceedings of the Royal Society A, Vol. 468, No. 2142, 2012, pp. 1716-1740. doi:10.1098/rspa.2011.0640 [14] P. Gaillard, “Wronskian Representation of Solutions of the NLS Equation and Higher Peregrine Breathers,” Journal of Mathematical Sciences: Advances and Applications, Vol. 13, No. 2, 2012, pp. 71-153.
[1] V. E. Zakharov, “Stability of Periodic Waves of Finite Amplitude on a Surface of a Deep Fluid,” Journal of Applied Mechanics and Technical Physics, Vol. 9, No. 2, 1968, pp. 86-94. [2] V. E. Zakharov and A. B. Shabat, “Exact Theory of Two Dimensional Self-Focusing and One Dimensinal Self-Modulation of Waves in Nonlinear Media,” Soviet Physics—JETP, Vol. 34, 1972, pp. 62-69. [3] A. R. Its and V. P. Kotlyarov, “Explicit Expressions for the Solutions of Nonlinear Schrodinger Equation,” Docklady Akademii Nauk SSSR, Vol. 965, No. 11, 1976, pp. 965-968 [4] D. Peregrine, “Water Waves, Nonlinear Schrodinger Equations and Their Solutions,” Journal of the Australian Mathematical Society, Vol. 25, 1983, pp. 16-43. [5] N. Akhmediev, V. Eleonsky and N. Kulagin, “Generation of Periodic Trains of Picosecond Pulses in an Optical Fiber: Exact Solutions,” Soviet Physics—JETP, Vol. 62, 1985, pp. 894-899. [6] N. Akhmediev, V. Eleonskii and N. Kulagin, “Exact First Order Solutions of the Nonlinear Schrodinger Equation,” Theoretical and Mathematical Physics, Vol. 72, No. 2, 1988, pp. 809-196. doi:10.1007/BF01017105 [7] N. Akhmediev, A. Ankiewicz and J. M. Soto-Crespo, “Rogue Waves and Rational Solutions of Nonlinear Schrodinger Equation,” Physical Review E, Vol. 80, 2009, Article ID: 026601. [8] D. J. Kedziora, A. Ankiewicz and N. Akhmediev, “Circular Rogue Wave Clusters,” Physical Review E, Vol. 84, 2011, Article ID: 056611. [9] P. Dubard, P. Gaillard, C. Klein and V. B. Matveev, “On Multi-Rogue Waves Solutions of the NLS Equation and Position Solutions of the KdV Equation,” European Physical Journal, Vol. 185, No. 1, 2010, pp. 247-258. doi:10.1140/epjst/e2010-01252-9 [10] P. Gaillard, “Families of Quasi-Rational Solutions of the NLS Equation and Multi-Rogue Waves,” Journal of Physics A: Mathematical and Theoretical, Vol. 44, No. 43, 2011, pp. 1-15. doi:10.1088/1751-8113/44/43/435204 [11] A. Ankiewicz, N. Akhmediev and P. A. Clarkson, “Rogue Waves, Rational Solutions, the Patterns of Their Zeros and Integral Relations,” Journal of Physics A: Mathematical and Theoretical, Vol. 43, 2010, pp. 1-9 [12] B. Guo, L. Ling and Q. P. Liu, “Nonlinear Schrodinger Equation: Generalized Darboux Transformation and Rogue Wave Solutions,” Physical Review E, Vol. 85, No. 2, 2012, Article ID: 026607. doi:10.1103/PhysRevE.85.026607 [13] Y. Ohta and J. Yang, “General High-Order Rogue Waves and Their Dynamics in the Nonlinear Schrodinger Equation,” Proceedings of the Royal Society A, Vol. 468, No. 2142, 2012, pp. 1716-1740. doi:10.1098/rspa.2011.0640 [14] P. Gaillard, “Wronskian Representation of Solutions of the NLS Equation and Higher Peregrine Breathers,” Journal of Mathematical Sciences: Advances and Applications, Vol. 13, No. 2, 2012, pp. 71-153.