Research Article
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Year 2022, , 24 - 34, 30.04.2022
https://doi.org/10.33187/jmsm.990955

Abstract

References

  • [1] A. Nakemura, R. Hirota, A new example of explode-decay solitary waves in one dimension, J. Phys. Soc. Jpn., 54(2) (1985), 491-499.
  • [2] A. N. W. Hone, Crum transformation and rational solutios of the non-focusing nonlinear Schr¨odinger equation, J. Phys. A: Math. Gen., 30(21) (1997), 7473-7483.
  • [3] S. Barran, M. Kovalyov, A note on slowly decaying solutions of the defocusing nonlinear Schrodinger equation, J. Phys. A: Math. Gen., 32(34) (1999), 6121.
  • [4] P. A. Clarkson, Special polynomials associated with rational solutions of defocusing nonlinear Schr¨odinger equation and fourth Pailev´e equation, European Journal of Applied Mathematics, 17 (2006), 293-322.
  • [5] J. Lenells, The defocusing nonlinear Schr¨odinger equation with t-periodic data new exact solutions, Nonlinear Analysis: Real World Applications, 25 (2015), 31-50.
  • [6] B. Prinari, F. Vitale, G. Biondini, Dark-bright soliton solutions with nontrivial polarization interactions to the three-component defocusing nonlinear Schrodinger equation with nonzero boudary conditions, Journal of Mathematical Physics, 56(7) (2015), 071505, 1-33.
  • [7] P. Gaillard, Families of quasi-rational solutions of the NLS equation and multi-rogue waves, Journal of Physics A, 44(43) (2011), 435204, 1-15.
  • [8] P. Gaillard, Wronskian representation of solutions of the NLS equation and higher Peregrine breathers, Journal of Mathematical Sciences: Advances and Applications, 13(2) (2012), 71-153.
  • [9] P. Gaillard, Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves, Journal of Mathematical Physics, 54 (2013), 013504, 1-32.
  • [10] 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, Journal of Physics A, 48(14) (2015), 145203, 1-23.
  • [11] P. Gaillard, Multi-parametric deformations of the Peregrine breather of order N solutions to the NLS equation and multi-rogue waves, Advances in Research, 4 (2015), 346-364.
  • [12] P. Gaillard, Towards a classification of the quasi rational solutions to the NLS equation, Theor. Math. Phys., 189 (2016), 1440-1449.
  • [13] P. Gaillard, Deformations of third order Peregrine breather solutions of the NLS equation with four parameters, Phys. Rev. E ., 88(4) (2013), 042903, 1-9.
  • [14] P. Gaillard, M. Gastineau, Families of deformations of the thirteenth Peregrine breather solutions to the NLS equation depending on twenty four parameters, J. Basic Appl. Res. Int., 21(3) (2017), 130-139.

Quasi-Rational and Rational Solutions to the Defocusing Nonlinear Schrödinger Equation

Year 2022, , 24 - 34, 30.04.2022
https://doi.org/10.33187/jmsm.990955

Abstract

Quasi-rational solutions to the defocusing nonlinear Schrödinger equation (dNLS) in terms of wronskians and Fredholm determinants of order $2N$ depending on $2N-2$ real parameters are given. We get families of quasi-rational solutions to the dNLS equation expressed as a quotient of two polynomials of degree $N(N+1)$ in the variables $x$ and $t$. We present also rational solutions as a quotient of determinants involving certain particular polynomials.

References

  • [1] A. Nakemura, R. Hirota, A new example of explode-decay solitary waves in one dimension, J. Phys. Soc. Jpn., 54(2) (1985), 491-499.
  • [2] A. N. W. Hone, Crum transformation and rational solutios of the non-focusing nonlinear Schr¨odinger equation, J. Phys. A: Math. Gen., 30(21) (1997), 7473-7483.
  • [3] S. Barran, M. Kovalyov, A note on slowly decaying solutions of the defocusing nonlinear Schrodinger equation, J. Phys. A: Math. Gen., 32(34) (1999), 6121.
  • [4] P. A. Clarkson, Special polynomials associated with rational solutions of defocusing nonlinear Schr¨odinger equation and fourth Pailev´e equation, European Journal of Applied Mathematics, 17 (2006), 293-322.
  • [5] J. Lenells, The defocusing nonlinear Schr¨odinger equation with t-periodic data new exact solutions, Nonlinear Analysis: Real World Applications, 25 (2015), 31-50.
  • [6] B. Prinari, F. Vitale, G. Biondini, Dark-bright soliton solutions with nontrivial polarization interactions to the three-component defocusing nonlinear Schrodinger equation with nonzero boudary conditions, Journal of Mathematical Physics, 56(7) (2015), 071505, 1-33.
  • [7] P. Gaillard, Families of quasi-rational solutions of the NLS equation and multi-rogue waves, Journal of Physics A, 44(43) (2011), 435204, 1-15.
  • [8] P. Gaillard, Wronskian representation of solutions of the NLS equation and higher Peregrine breathers, Journal of Mathematical Sciences: Advances and Applications, 13(2) (2012), 71-153.
  • [9] P. Gaillard, Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves, Journal of Mathematical Physics, 54 (2013), 013504, 1-32.
  • [10] 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, Journal of Physics A, 48(14) (2015), 145203, 1-23.
  • [11] P. Gaillard, Multi-parametric deformations of the Peregrine breather of order N solutions to the NLS equation and multi-rogue waves, Advances in Research, 4 (2015), 346-364.
  • [12] P. Gaillard, Towards a classification of the quasi rational solutions to the NLS equation, Theor. Math. Phys., 189 (2016), 1440-1449.
  • [13] P. Gaillard, Deformations of third order Peregrine breather solutions of the NLS equation with four parameters, Phys. Rev. E ., 88(4) (2013), 042903, 1-9.
  • [14] P. Gaillard, M. Gastineau, Families of deformations of the thirteenth Peregrine breather solutions to the NLS equation depending on twenty four parameters, J. Basic Appl. Res. Int., 21(3) (2017), 130-139.
There are 14 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Pierre Gaillard 0000-0002-7073-8284

Publication Date April 30, 2022
Submission Date September 3, 2021
Acceptance Date January 19, 2022
Published in Issue Year 2022

Cite

APA Gaillard, P. (2022). Quasi-Rational and Rational Solutions to the Defocusing Nonlinear Schrödinger Equation. Journal of Mathematical Sciences and Modelling, 5(1), 24-34. https://doi.org/10.33187/jmsm.990955
AMA Gaillard P. Quasi-Rational and Rational Solutions to the Defocusing Nonlinear Schrödinger Equation. Journal of Mathematical Sciences and Modelling. April 2022;5(1):24-34. doi:10.33187/jmsm.990955
Chicago Gaillard, Pierre. “Quasi-Rational and Rational Solutions to the Defocusing Nonlinear Schrödinger Equation”. Journal of Mathematical Sciences and Modelling 5, no. 1 (April 2022): 24-34. https://doi.org/10.33187/jmsm.990955.
EndNote Gaillard P (April 1, 2022) Quasi-Rational and Rational Solutions to the Defocusing Nonlinear Schrödinger Equation. Journal of Mathematical Sciences and Modelling 5 1 24–34.
IEEE P. Gaillard, “Quasi-Rational and Rational Solutions to the Defocusing Nonlinear Schrödinger Equation”, Journal of Mathematical Sciences and Modelling, vol. 5, no. 1, pp. 24–34, 2022, doi: 10.33187/jmsm.990955.
ISNAD Gaillard, Pierre. “Quasi-Rational and Rational Solutions to the Defocusing Nonlinear Schrödinger Equation”. Journal of Mathematical Sciences and Modelling 5/1 (April 2022), 24-34. https://doi.org/10.33187/jmsm.990955.
JAMA Gaillard P. Quasi-Rational and Rational Solutions to the Defocusing Nonlinear Schrödinger Equation. Journal of Mathematical Sciences and Modelling. 2022;5:24–34.
MLA Gaillard, Pierre. “Quasi-Rational and Rational Solutions to the Defocusing Nonlinear Schrödinger Equation”. Journal of Mathematical Sciences and Modelling, vol. 5, no. 1, 2022, pp. 24-34, doi:10.33187/jmsm.990955.
Vancouver Gaillard P. Quasi-Rational and Rational Solutions to the Defocusing Nonlinear Schrödinger Equation. Journal of Mathematical Sciences and Modelling. 2022;5(1):24-3.

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