Breather Solutions in NLS

• Breather solutions are exact, time-periodic and spatially localized waveforms derived from the nonlinear Schrödinger equation that model modulation instability and rogue events.
• The displaced phase–amplitude variable method reduces the NLS to a Riccati-type equation, yielding explicit analytic expressions for SFB, Ma, and rational breathers.
• Amplitude amplification factors and observed wavefront dislocations provide practical diagnostics for extreme wave events in hydrodynamics, optics, and plasma physics.

Breather solutions are exact, time-periodic, spatially localized solutions to nonlinear dispersive PDEs, fundamental in describing modulation instability, envelope dynamics, and rogue phenomena in physical systems modeled by the nonlinear Schrödinger (NLS) equation. The explicit connection between different breather solutions—Soliton on Finite Background (SFB), Ma breather, and the rational breather—can be rigorously established via the displaced phase–amplitude variable method, which directly reduces the NLS to a Riccati-type equation for the modulated amplitude. This approach provides constructive, analytic formulae for all three solutions, clarifies their parameter continuations, and establishes the precise regimes in the parameter space where each type appears, including their amplitude amplification factors. The displaced phase–amplitude representation also elucidates physical phenomena such as wavefront dislocation and phase singularities at vanishing amplitude, as observed in experiments. These results have significant implications for the prediction, synthesis, and diagnosis of extreme wave events in hydrodynamics, optics, and related fields.

1. Displaced Phase–Amplitude Variable Method

A central theme is the analytic construction of breather solutions to the spatial NLS equation,

$$\partial_\xi \psi + i \beta \partial_\tau^2 \psi + i \gamma |\psi|^2 \psi = 0,$$

using a displaced phase–amplitude ansatz. The method decomposes the field amplitude as

$$A(\xi,\tau) = A_0(\xi) [G(\xi,\tau) e^{i \phi(\xi)}-1],$$

where the background plane wave is

$A_0(\xi) = r_0 e^{-i \gamma r_0^2 \xi},$

and all temporal dependence is shifted to the displaced amplitude $G$, with a spatially varying phase $\phi(\xi)$.

Critically, φ is required to depend on space only (temporally independent). Substituting the ansatz into the NLS and separating real and imaginary parts yields two coupled ODEs. By specific linear combinations, these reduce to a nonlinear Riccati-type equation for $G$:

$$\partial_\xi G + \gamma r_0^2 \sin(2\phi) G - \gamma r_0^2 \sin(\phi) G^2 = 0.$$

The substitution $G = 1/H$ converts this to a linear ODE for $H$, solvable via integrating factors. The general solution is

$$G(\phi, \tau) = \frac{P(\phi)}{Q(\phi) - \zeta(\tau)}$$

with $P(\phi)$, $Q(\phi)$ from spatial integration and $\zeta(\tau)$ an a priori arbitrary function determined by temporal evolution. Temporal evolution is matched by considering a nonlinear oscillator equation for $G$, constraining $\zeta(\tau)$ and resulting in the three principal breather families.

2. Explicit Forms of Breather Solutions and Amplitude Amplification Factors

By selecting distinct forms of $\zeta(\tau)$, the three canonical breather solutions are derived:

Breather Type	ζ(τ)	Key Parameters	Amplification Factor	Regime/Notes
Soliton on Finite Background	cos(ντ)	$\hat{\nu} \equiv \nu/(r_0\sqrt{\gamma/\beta})$ < $\sqrt{2}$	$1+\sqrt{4-2\hat{\nu}^2}$, $1<AAF_S<3$	Satisfies periodicity in τ
Ma Breather	cosh(μτ)	$\hat{\mu} \equiv \mu/(r_0\sqrt{\gamma/\beta})$ > 0	$1+\sqrt{4+2\hat{\mu}^2}$, $AAF_M>3$	Localized in ξ, periodic in τ
Rational Breather	$1 - \frac{1}{2}\nu^2\tau^2$ (or $1 + \frac{1}{2}\mu^2\tau^2$ )	Parameter-free limit	3	Limit ν→0 or μ→0

For each, the full NLS solution takes the form

$$A(\xi, \tau) = A_0(\xi) \times \mathrm{BreatherFactor}(\xi, \tau)$$

with explicit expressions for $\mathrm{BreatherFactor}$ from the solution for $G$. The amplitude amplification factor (AAF) quantifies the excess of peak amplitude over the background and is central to rogue wave criteria: the rational (Peregrine) case yields the sharp value $AAF_R = 3$, while the SFB and Ma families interpolate up to or above this threshold, delineating parameter-space zones of extremality.

3. Structural Relations Between NLS Breathers

The analytic continuation of the temporal parameter in $\zeta(\tau)$ connects the three breather types:

• The SFB is analytically continued to the Ma breather by substituting $\nu = i\mu$, which transforms $\cos(\nu\tau) \rightarrow \cosh(\mu\tau)$.
• In the limit $\nu \to 0$ (or $\mu \to 0$), both SFB and Ma breathers reduce to the rational/Peregrine breather.
• The AAF for SFB and Ma converges to 3 in this limit: $AAF_S(\nu \to 0) = AAF_M(\mu \to 0) = 3$.

Graphical representations (“SegitigaBreather”) clarify these connections, placing all three solutions as endpoints or analytic continuations within a unified geometric scheme in parameter space.

4. Physical Interpretation: Wavefield Structure, Dislocations, and Singularities

Reconstruction into the physical field through modulation with a carrier wave,

$$\eta(x, t) = A(\xi, \tau) e^{i(k_0 x - \omega_0 t)} + c.c.,$$

with co-moving coordinates $\xi = x$, $\tau = t - x/V_0$, produces envelope wave packets exhibiting the characteristic periodic focusing-defocusing cycles—the "breathing".

A key phenomenon is wavefront dislocation, at which wave crests merge or split—these occur at spatiotemporal loci where the envelope amplitude vanishes. Such points correspond to phase singularities, with the physical phase being undefined (ill-posed), providing a diagnostic experimental signature of breathers and associated with energy localization and modulational instability (e.g., the Benjamin–Feir instability regime).

5. Applications and Implications in Physical Systems

The displaced phase–amplitude framework and resultant breather solutions have wide multi-disciplinary relevance:

• Hydrodynamics: The SFB models homoclinic orbits linked to the Benjamin–Feir instability, directly relevant for the generation and prediction of rogue oceanic waves. In laboratory tanks, these breathers determine the parameter regimes for extreme wave observation.
• Nonlinear Optics/Plasma Physics: The Ma and SFB breathers correspond to modulated pulses in optical fibers and plasma, with parameters (γ, β, r₀) tailored to the physical dispersive and nonlinear coefficients.
• Extreme Events Diagnosis: The amplitude amplification factor (AAF) gives a quantitative metric for identifying potential for rogue wave events; a value ≥3 is commonly regarded as indicative of “rogue” behavior.
• Experimental Diagnostics: The predicted wavefront dislocations and phase singularities provide signatures for diagnostics (e.g., spatiotemporal interferometry) when attempting to correlate theoretical breather solutions with measured data.

6. Mathematical and Theoretical Significance

The construction highlights the utility of variable separation and reduction to Riccati (and then linear) dynamic forms for explicitly deriving large classes of physically meaningful exact solutions. The precise classification of the solution families and specification of their parameter dependence establishes a template for generalizations—including to variable coefficient, inhomogeneous, or multi-dimensional NLS variants.

Moreover, the programmatic reduction of the nonlinear stability properties and phase propagation behavior to properties of this analytic ansatz explicitly connects the theory of integrable systems to observable, extreme localization phenomena.

7. Summary Table: Core Features of NLS Breather Types

Type	ζ(τ)	Form	Peak Amplitude	Reduction/Limit	Physical Regime
SFB	$\cos(\nu\tau)$	SFB analytic solution	$1 + \sqrt{4-2\hat{\nu}^2}$	ν→0 → Rational	Homoclinic, finite background
Ma Breather	$\cosh(\mu\tau)$	Ma analytic solution	$1 + \sqrt{4+2\hat{\mu}^2}$	μ→0 → Rational	Spatially localized, time-periodic
Rational Breather	$1 - \frac{1}{2}\nu^2\tau^2$ or $1 + \frac{1}{2}\mu^2\tau^2$	Rational analytic solution	3	Endpoint of SFB/Ma	Maximum focusing, Peregrine regime

References to Related Research

This displaced phase–amplitude methodology and the resulting explicit connection of the canonical NLS breather solutions were explicitly developed in (Karjanto et al., 2011). The completeness of the construction—encompassing the full moduli space of classical breathers, clarifying interrelations, and grounding experimentally significant diagnostics (AAF, dislocation, singularities)—establishes a rigorous and practical foundation for both the mathematical theory of integrable PDEs and the applied modeling of nonlinear wave phenomena.