WIAS Preprint No. 2208, (2016)

Infinite hierarchy of nonlinear Schrödinger equations and their solutions


  • Ankiewicz, Adrian
  • Kedziora, David Jacob
  • Chowdury, Amdad
  • Bandelow, Uwe
    ORCID: 0000-0003-3677-2347
  • Akhmediev, Nail

2010 Mathematics Subject Classification

  • 35Q55 37K10 35C08

2010 Physics and Astronomy Classification Scheme

  • 05.45.Yv, 42.65.Tg, 42.81.qb


  • Nonlinear Schrödinger Equations, Infinite Hierarchy, Solitons, Breathers, Rogue Waves




We study the infinite integrable nonlinear Schrödinger equation (NLSE) hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, AB breathers, Kuznetsov-Ma breathers, periodic solutions and rogue wave solutions for this infinite order hierarchy. We find that 'even' order equations in the set affect phase and 'stretching factors' in the solutions, while 'odd' order equations affect the velocities. Hence 'odd' order equation solutions can be real functions, while 'even' order equation solutions are always complex.

Appeared in

  • Phys. Rev. E, 93 (2016) pp. 012206/1--012206/10.

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