Differential Relations for the Solutions to the NLS Equation and Their Different Representations

Solutions to the focusing nonlinear Schr\"odinger equation (NLS) of order $N$ depending on $2N-2$ real parameters in terms of wronskians and Fredholm determinants are given. These solutions give families of quasi-rational solutions to the NLS equation denoted by $v_{N}$ and have been explicitly constructed until order $N = 13$. These solutions appear as deformations of the Peregrine breather $P_{N}$ as they can be obtained when all parameters are equal to $0$. These quasi rational solutions can be expressed as a quotient of two polynomials of degree $N(N+1)$ in the variables $x$ and $t$ and the maximum of the modulus of the Peregrine breather of order $N$ is equal to $2N+1$. \\ Here we give some relations between solutions to this equation. In particular, we present a connection between the modulus of these solutions and the denominator part of their rational expressions. Some relations between numerator and denominator of the Peregrine breather are presented.

Keywords:

NLS equation, Fredholm determinants, NLS equation, Peregrine breathers, Rogue waves, Wronskians Wronskians,

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[1] V. E. Zakharov, Stability of periodic waves of finite amplitude on a surface of a deep fluid, J. Appl. Tech. Phys., 9 (1968), 86-94.
[2] V. E. Zakharov, A. B. Shabat Exact theory of two dimensional self focusing and one dimensinal self modulation of waves in nonlinear media, Sov. Phys. JETP, 34 (1972), 62-69.
[3] A. R. Its, A. V. Rybin, M. A. Salle, Exact integration of nonlinear Schrödinger equation, Teore. Mat. Fiz., 74 (1988), N. 1, 29-45.
[4] A. R. Its, V.P. Kotlyarov, Explicit expressions for the solutions of nonlinear Schrödinger equation, Dockl. Akad. Nauk. SSSR, S. A, 965(11) (1976).
[5] Y. C. Ma, The perturbed plane-wave solutions of the cubic nonlinear Schrödinger equation, Stud. Appl. Math. 60 (1979), 43-58.
[6] D. H. Peregrine, Water waves, nonlinear Schr¨odinger equations and their solutions, J. Austral. Math. Soc. Ser. B, 25 (1983), 16-43.
[7] N. Akhmediev, V. Eleonskii, N. Kulagin, Exact first order solutions of the nonlinear Schr¨odinger equation, Th. Math. Phys., 72(2) (1987), 183-196.
[8] N. Akhmediev, V. Eleonsky, N. Kulagin, Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions, Sov. Phys. J.E.T.P., 62 (1985), 894-899.
[9] V. Eleonskii, I. Krichever, N. Kulagin, Rational multisoliton solutions to the NLS equation, Soviet Doklady 1986 sect. Math. Phys., 287 (1986), 606-610.
[10] N. Akhmediev, A. Ankiewicz, J. M. Soto-Crespo, Rogue waves and rational solutions of nonlinear Schr¨odinger equation, Physical Review E,80( 026601) (2009).
[11] N. Akhmediev, A. Ankiewicz, P. A. Clarkson, Rogue waves, rational solutions, the patterns of their zeros and integral relations, J. Phys. A : Math. Theor., 43(122002) (2010), 1-9.
[12] N. Akhmediev, A. Ankiewicz, D. J. Kedziora, Circular rogue wave clusters, Phys. Review E, 84 (2011), 1-7.
[13] A. Chabchoub, H. Hoffmann, M. Onorato, N. Akhmediev, Super rogue waves: Observation of a higher-order breather in water waves, Phys. Review X, 2 (2012), 1-6.
[14] P. Dubard, P. Gaillard, C. Klein, V. B. Matveev, On multi-rogue waves solutions of the NLS equation and positon solutions of the KdV equation, Eur. Phys. J. Special Topics, 185 (2010), 247-258.
[15] P. Dubard, V. B. Matveev, Multi-rogue waves solutions : from the NLS to the KP-I equation, Nonlinearity, 26 (2013), 93-125.
[16] P. Gaillard, Families of quasi-rational solutions of the NLS equation and multi-rogue waves, J. Phys. A : Meth. Theor., 44 (2011), 1-15.
[17] P. Gaillard, Wronskian representation of solutions of the NLS equation and higher Peregrine breathers, Jour. Of Math. Sciences : Advances and Applications, 13(2) (2012), 71-153.
[18] P. Gaillard, Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves, Jour. Of Math. Phys., 54 (2013), 013504-1-32.
[19] D.J. Kedziora, A. Ankiewicz, N. Akhmediev, Triangular rogue wave cascades, Phys. Rev. E, 86(056602) (2012), 1-9.
[20] D. J. Kedziora, A. Ankiewicz, N. Akhmediev, Circular rogue wave clusters, Phys. Review E, 84 (2011), 056611-1-7.
[21] D. J. Kedziora, A. Ankiewicz, N. Akhmediev, Classifying the hierarchy of the nonlinear Schr¨odinger equation rogue waves solutions, Phys. Review E, 88 (2013), 013207-1-12.
[22] B. Guo, L. Ling, Q.P. Liu, Nonlinear Schr¨odinger equation: Generalized Darboux transformation and rogue wave solutions, Phys. Rev. E, 85 (2012), 026607.
[23] Y. Ohta, J. Yang, General high-order rogue waves and their dynamics in the nonlinear Schr¨odinger equation, Pro. R. Soc. A, 468 (2012), 1716-1740.
[24] A.O. Smirnov, Solution of a nonlinear Schr¨odinger equation in the form of two phase freak waves, Theor. Math. Phys., 173 (2012), 1403-1416.
[25] L. Ling, L.C. Zhao, Simple determinant representation for rogue waves of the nonlinear Schr¨odinger equation, Phys. Rev. E, 88 (2013), 043201-1-9.
[26] P. Gaillard, Other 2N-2 parameters solutions to the NLS equation and 2N+1 highest amplitude of the modulus of the N-th order AP breather, J. Phys. A: Math. Theor., 48 (2015), 145203-1-23.
[27] P. Gaillard, Multi-parametric deformations of the Peregrine breather of order N solutions to the NLS equation and multi-rogue waves, Adv. Res., 4 (2015), 346-364.
[28] P. Gaillard, Deformations of third order Peregrine breather solutions of the NLS equation with four parameters, Phys. Rev. E, 88 (2013), 042903-1-9.
[29] P. Gaillard, M. Gastineau Twenty parameters families of solutions to the NLS equation and the eleventh Peregrine breather, Commun. Theor. Phys, 65 (2016), 136-144.
[30] P. Gaillard, M. Gastineau Twenty two parameters deformations of the twelfth Peregrine breather solutions to the NLS equation, Adv. Res., 10 (2016), 83-89.
[31] P. Gaillard, Towards a classification of the quasi rational solutions to the NLS equation, Theor. And Math. Phys., 189 (2016), 1440-1449.
[32] P. Gaillard, M. Gastineau Families of deformations of the thirteenth Peregrine breather solutions to the NLS equation depending on twenty four parameters, J. Bas. Appl. Res. Int., 21(3) (2017), 130-139.
[33] A.P. Vorob’ev, On the rational solutions of the second Painlev´e equation, Differ. Uravn., 1(1) (1965), 79-81.
[34] A.I. Yablonskii, On rational solutions of the second Painlev´e equation, Vesti AN BSSR, Ser. Fiz.-Tech. Nauk, (3) (1959), 30-35.