[Paper Review] Derivation of the NLS breather solutions using displaced phase-amplitude variables

N. Karjanto, E. van Groesen|arXiv (Cornell University)|Oct 21, 2011

Nonlinear Waves and Solitons24 references11 citations

TL;DR

This paper derives three breather solutions of the nonlinear Schrödinger equation—Soliton on Finite Background (SFB), Ma breather, and rational breather—using a displaced phase-amplitude variable formulation with time-independent phase. The method reduces the dynamics to a nonlinear oscillator in a position-dependent potential, yielding explicit solutions that unify the three breathers through a common framework and reveal wavefront dislocations and phase singularities in physical wave fields.

ABSTRACT

Breather solutions of the nonlinear Schrödinger equation are derived in this paper: the Soliton on Finite Background, the Ma breather and the rational breather. A special Ansatz of a displaced phase-amplitude equation with respect to a background is used as has been proposed by van Groesen et. al. (2006). Requiring the displaced phase to be temporally independent, has as consequence that the dynamics at each position is described by the motion of a nonlinear autonomous oscillator in a potential energy that depends on the phase and on the spatial phase change. The relation among the breather solutions is confirmed by explicit expressions, and illustrated with the amplitude amplification factor. Additionally, the corresponding physical wave field is also studied and wavefront dislocation together with phase singularity at vanishing amplitude are observed in all three cases.

Motivation & Objective

• To derive exact breather solutions of the nonlinear Schrödinger equation using a displaced phase-amplitude formulation.
• To unify the Soliton on Finite Background (SFB), Ma breather, and rational breather under a single analytical framework.
• To investigate the physical wave field behavior, particularly wavefront dislocations and phase singularities at vanishing amplitude.
• To clarify the mathematical and physical relationships among the three breather types through explicit parameterization and amplitude amplification factors.
• To demonstrate the relevance of these solutions for modeling extreme wave events in water waves and nonlinear optics.

Proposed method

• Employ a displaced phase-amplitude Ansatz: $ A( heta, au) = A_0( heta)[G( heta, au)e^{i ilde{ heta}( heta)} - 1] $, where $ ilde{ heta} $ is spatially dependent and time-independent.
• Derive a Riccati-like equation for the amplitude envelope $ G $, which is transformed into a linear first-order ODE for $ H = 1/G $ using an integrating factor $ ilde{P}( heta) $.
• Solve the resulting ODE to obtain $ H( heta, au) = [ ilde{Q}( heta) - ilde{ ho}( au)] / ilde{P}( heta) $, with $ ilde{ ho}( au) $ as a time-dependent integration constant.
• Apply the condition that $ ilde{ heta}( heta) $ is invertible to reparameterize in terms of phase, enabling explicit construction of the three breather solutions.
• Use the Argand diagram to visualize the time evolution of the complex amplitude, showing periodic elliptical or linear trajectories depending on the breather type.
• Construct physical wave fields via $ ilde{ ho}(x,t) = A( heta, au)e^{i(k_0x - ilde{ u}t)} + ext{c.c.} $, and analyze density plots to identify wavefront dislocations and phase singularities.

Experimental results

Research questions

• RQ1 How can the three distinct breather solutions of the NLS equation be systematically derived from a unified phase-amplitude formulation?
• RQ2 What is the role of a time-independent displaced phase in generating periodic, localized, and rational breather solutions?
• RQ3 How do the amplitude amplification factors and wavefront dislocations differ across the SFB, Ma breather, and rational breather?
• RQ4 In what way do wavefront dislocations and phase singularities emerge in the physical wave fields of these breathers?
• RQ5 What is the limiting relationship between the SFB, Ma breather, and rational breather as parameters $ \tilde{\nu} \to 0 $ and $ \tilde{\mu} \to 0 $?

Key findings

• The SFB, Ma breather, and rational breather are all derived from the same displaced phase-amplitude framework with a time-independent phase, confirming their unification under one analytical approach.
• The amplitude amplification factor for the SFB is $ 2r_0 $ at $ \tau = 0 $, while for the Ma breather it reaches $ 2r_0 $ at $ \xi = 0 $, and for the rational breather it is unbounded at $ \xi = \tau = 0 $, indicating extreme localization.
• In the Argand diagram, the SFB exhibits elliptical trajectories centered at $ (-1, 0) $, the Ma breather shows radial lines from $ (-1, 0) $, and the rational breather approaches a point at $ (-1, 0) $ in the limit $ \tilde{\nu}, \tilde{\mu} \to 0 $.
• All three breather solutions exhibit wavefront dislocation in their physical wave fields, characterized by splitting or merging of wave crests, which occurs when the Chu-Mei quotient diverges at vanishing amplitude.
• Phase singularities—points of undefined phase—occur at locations where the wave amplitude vanishes, a feature consistently observed across all three breather types.
• The rational breather emerges as the limiting case of both the SFB and Ma breather when $ \tilde{\nu} \to 0 $ and $ \tilde{\mu} \to 0 $, respectively, confirming their hierarchical relationship.