Multi-breathers and high order rogue waves for the nonlinear Schrödinger equation on the elliptic function background

Abstract

We construct the multi-breather solutions of the focusing nonlinear Schrödinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and non-trivial phase-modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multi-breather solutions with different velocities in the limit $t\to \pm \infty$, where the solution in the neighborhood of each breather tends to the simple one-breather solution. Furthermore, we exactly solve the linearized NLSE using the squared eigenfunction and determine the unstable spectra for elliptic function background. By using them, the Akhmediev breathers arising from these modulational instabilities are plotted and their dynamics are revealed. Finally, we provide the rogue-wave and higher-order rogue-wave solutions by taking the special limit of the breather solutions at branch points and the generalized Darboux transformation. The resulting dynamics of the rogue waves involves rich phenomena: depending on the choice of the background and possessing different velocities relative to the background. We also provide an example of the multi- and higher-order rogue wave solution.

45 pages, 16 figures

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• Stability of elliptic function solutions for the focusing modified KdV equation
• Localized structures on librational and rotational travelling waves in the sine-Gordon equation
• The multi elliptic-localized solutions and their asymptotic behaviors for the mKdV equation
• Broader Universality of Rogue Waves of Infinite Order
• The solutions of classical and nonlocal nonlinear Schrödinger equations with nonzero backgrounds: Bilinearisation and reduction approach
• Current production in ring condensates with a weak link
• Measuring the rogue wave pattern triggered from Gaussian perturbations by deep learning
• Tsunami Solitons Emerging from Superconducting Gap
• Rogue waves on the periodic wave background in the focusing nonlinear Schrodinger equation