Wronskian Representation of Solutions of NLS Equation, and Seventh Order Rogue Wave

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,"

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.