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Spectral problem for the complex mKdV equation: singular manifold method and Lie symmetries

Paz Albares, Pilar G. Estévez, Alejandro González-Parra, Paula del Olmo

Abstract

This article addresses the study of the complex version of the modified Korteweg-de Vries equation using two different approaches. Firstly, the singular manifold method is applied in order to obtain the associated spectral problem, binary Darboux transformations and $τ$-functions. The second part concerns the identification of the classical Lie symmetries for the spectral problem. The similarity reductions associated to these symmetries allow us to derive the reduced spectral problems and first integrals for the ordinary differential equations arising from such reductions.

Spectral problem for the complex mKdV equation: singular manifold method and Lie symmetries

Abstract

This article addresses the study of the complex version of the modified Korteweg-de Vries equation using two different approaches. Firstly, the singular manifold method is applied in order to obtain the associated spectral problem, binary Darboux transformations and -functions. The second part concerns the identification of the classical Lie symmetries for the spectral problem. The similarity reductions associated to these symmetries allow us to derive the reduced spectral problems and first integrals for the ordinary differential equations arising from such reductions.
Paper Structure (16 sections, 58 equations, 3 figures, 1 table)

This paper contains 16 sections, 58 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Singular manifold method
  3. Truncated expansion
  4. Spectral problem
  5. Darboux transformations
  6. Iteration of solutions
  7. Soliton solutions
  8. Line solitons
  9. Breathers
  10. Rogue waves
  11. Lie symmetries
  12. Lie algebra
  13. Similarity reductions
  14. $k_1\neq 0$
  15. $k_1=0,\,k_2\neq 0,\,k_3\neq 0$
  16. ...and 1 more sections

Figures (3)

  • Figure 1: One-soliton solution (left) and two-soliton solution (right) for $k_0=0,j_0=1,d_1=1,d_2=2,\theta_1=0.1,\theta_2=0.2.$
  • Figure 2: Analogous Akhmediev breather (left) and Kuznetsov-Ma breather (right) for the complex mKdV equation, with parameter $k_0=1,j_0=1,\theta_1=0.2,\varphi_1=0.2$.
  • Figure 3: Rogue waves for the complex mKdV equation when $a=0,z_0=1$ (left) and $a\neq 0,z_0=0$ (right), for parameters $k_0=1,j_0=1,a_0=0,b_0=0$.