Stokes Soliton Phenomena

Updated 30 July 2025

Stokes soliton is a solitary wave solution defined by the activation of exponentially small contributions via the Stokes phenomenon across complex domains.
In nonlinear optics, Stokes solitons form through Raman interactions and mode-locking in microcavities, yielding dual phase-locked states for advanced photonic applications.
They also emerge in fluid dynamics and integrable systems where variable geometries and spectral background modes shape robust, coherent wave states with practical implications.

Stokes soliton refers to a class of solitary and quasi-solitary wave solutions whose structure, formation, or stability is governed by Stokes phenomena—specifically, the switching-on of exponentially small, beyond-all-orders wave contributions—and to related coherent states in nonlinear optics, hydrodynamics, and statistical spectral theory linked to Stokes background modes. The term encompasses several distinct but interrelated physical and mathematical contexts: exponential asymptotics and smoothing of Stokes lines in high-order dispersive equations; Raman-induced solitonic states in optical microcavities; generalizations in hydrodynamics over variable depth; and spectral/statistical soliton dynamics dominated by Stokes (background) modes or bands.

1. Exponential Asymptotics and the Stokes Phenomenon in Solitary Wave Theory

Stokes solitons in the context of higher-order dispersive wave equations, such as the fifth-order Korteweg–de Vries (5KdV) equation, arise due to the interplay between divergent asymptotic expansions and the activation of hidden exponentially small oscillatory tails as a result of Stokes lines crossing in the complexified physical domain (Trinh, 2014). The formal construction begins with an asymptotic series for the wave profile,

$u(x) = \sum_{n=0}^{\infty} \varepsilon^{2n} u_n(x),$

which is typically divergent due to the presence of complex-plane singularities in the leading-order solution. The divergence is controlled by the behavior

$u_n \sim \frac{Q(z)\,\Gamma(2n+\gamma)}{\chi(z)^{2n+\gamma}}, \quad n \to \infty$

where $\chi(z)$ vanishes at these singular points.

The "Stokes line" is determined by the locus where

$\operatorname{Im}(-\chi^2) = 0,\quad \operatorname{Re}(-\chi^2) \geq 0,$

so that as one traverses such a line in the complexified $x$ -plane, an exponentially small, non-analytic contribution to the solution becomes "switched on." The mechanism of Stokes line smoothing is mathematically encoded by analyzing the optimally-truncated asymptotic series remainder via rescaling near the Stokes line, which leads to an error-function type smoothing of the jump—delineating the Stokes phenomenon.

The resulting total solution thus includes an exponentially small oscillatory tail of the form

$u_\text{exp} \sim -\frac{2\Lambda\pi}{\varepsilon^2} e^{-\pi/(2\gamma \varepsilon)} \sin(x/\varepsilon)$

in the real domain. This result rigorously demonstrates that strictly localized "classical" solitary waves (without oscillatory tails) do not exist for such higher-order dispersive equations: every generalized solitary wave bears a hidden oscillatory Stokes soliton tail, switched on across the Stokes line.

2. Stokes Solitons in Optical Microcavities via Raman Interactions

In nonlinear optical systems, specifically in whispering gallery mode microcavities, the Stokes soliton denotes a soliton mode arising not from conventional Kerr nonlinearity alone but through a Raman-mediated process enabled by the interaction of two distinct transverse mode families (Yang et al., 2016). The essential features:

Formation Mechanism: A primary Kerr soliton generates a Raman gain spectrum, which preferentially amplifies noise in a red-shifted, nearly frequency-commensurate secondary transverse mode. When the primary soliton power exceeds a threshold,

$P_{\mathrm{th}} = \frac{\kappa_p^{\text{ext}} \kappa_s}{2R}\left(1+\frac{1}{2\gamma}\right),$

the Stokes soliton is formed and locked to the repetition rate of the primary.

Raman Locking and Trapping: The Stokes soliton and the primary soliton become phase-locked due to cross-phase modulation, and their spatial overlap allows a new form of soliton trapping through both Raman gain and induced optical potentials.
Distinct Transverse Mode Families: The two solitons participate in Raman interaction only if their respective free spectral ranges (FSR) are closely matched, allowing phase and energy transfer efficiency.
Applications: These dual, locked soliton states, spectrally separated and yet phase coherent, enable new routes for difference frequency generation, mid-IR sources, and high-resolution photonic devices, expanding the traditional landscape of microcavity soliton dynamics.

The Stokes soliton in this context is therefore not a byproduct of fission or supercontinuum generation but is actively seeded and regenerated by the Raman gain initiated by the primary soliton, yielding a distinctive, mode-selective solitary state.

3. Soliton Dynamics Over Uneven Bottoms and the KdV2 Equation

In geophysical fluid dynamics, the Stokes soliton also appears in connection to solitary wave solutions of extended Korteweg–de Vries (KdV2) equations over variable topography (Rowlands et al., 2016). For an uneven riverbed or channel, solitons adapt their amplitude, width, and phase according to the local depth modification, governed by a two-scale reductive perturbation analysis.

The procedure involves rewriting the KdV2 equation with coefficients dependent on a slowly-varying bottom profile $h(x)$ , then mapping the variable-coefficient equation to an effective constant-coefficient equation via a stretching transformation,

$y = \int_0^x a(\epsilon x)dx - t, \quad \zeta = y/a(x_1), \quad h = h(\epsilon x),$

yielding a modified soliton profile of the form

$\eta_0 = \bar{A} \, \mathrm{sech}^2 (\bar{B}\zeta), \quad \bar{A} = A(1+q\delta h), \quad \bar{B} = B(1+\tfrac{q}{2}\delta h).$

As the soliton traverses regions of varying depth, its phase accumulates a shift

$\Delta \Phi \approx \frac{(\beta + 1/2)}{v} \int_{L_1}^{L_2} \delta h(x)\,dx,$

adapting smoothly, but with possible irreversible radiation emission not captured by first-order approximations. This indicates robust, physically observable Stokes-type soliton behavior in natural, inhomogeneous media.

4. Spectral Theory, Stokes Modes, and Soliton Gases

In the spectral kinetic theory of integrable systems, particularly for the focusing nonlinear Schrödinger equation (fNLS), the "Stokes mode" refers to an exceptional spectral band corresponding to a persistent nonzero background (plane wave) (El et al., 2019). When this mode is present, the resulting spectral state is a "breather gas"—localized soliton-like excitations on a finite background. If the Stokes spectral band is collapsed (background removed), one recovers a soliton gas.

Key features include:

Spectral Structure: The exceptional Stokes band (denoted $_0$ ) appears in the spectral data as a non-vanishing cut in the complex plane.
Nonlinear Dispersion Relations: The presence or absence of the Stokes mode modifies the nonlinear group velocity of solitons and breathers:

$s_0(\eta) = -2\frac{\mathrm{Im}[\eta R_0(\eta)]}{\mathrm{Im} R_0(\eta)}$

with $R_0(z) = \sqrt{z^2-\delta_0^2}$ , where $\delta_0$ encodes the Stokes band width.

Kinetic Equations: The mean field and interaction of soliton gases are described by kinetic equations incorporating the effect of the background Stokes mode:

$\partial_t u + \partial_x (u s) = 0, \quad s(\eta) = s_0(\eta) + \int \Delta(\eta,\mu)[s(\eta)-s(\mu)] u(\mu)|d\mu|$

In regimes where the Stokes mode persists, new collective states such as soliton (or breather) condensates emerge, with explicit density of state formulae connected to the background intensity. These structures have direct implications for rogue wave generation and energy transport in nonlinear optics and oceanography.

5. Stokes Solitons in Nonlinear Optics and Dissipative Systems

In dissipative optical fiber systems, the concept of a Stokes soliton (or Raman dissipative soliton) is important for mode-locked lasers operating under normal dispersion (Podivilov et al., 2016). Coherent Raman feedback ensures the synchronous generation of a Stokes-shifted dissipative soliton, circumventing the fundamental energy limits imposed by spontaneous Raman scattering.

The key workflow involves:

Generation of a primary dissipative soliton (DS) in a normal-dispersion laser;
Intracavity feedback for the Stokes-shifted wave ensures synchronous, phase-locked Raman dissipative soliton (RDS);
Mixing these DS and RDS pulses in a highly nonlinear fiber produces new spectral satellites via cascaded four-wave mixing (FWM), forming a "dissipative soliton comb":

$B_\text{out}(t) = B(t)\exp[2i\gamma L I(t)(1+\cos(\Delta\omega_{ST} t))]$

with pulse power spectra governed by Bessel functions,

$P(\omega_n) = P_0 |J_n(2\gamma L I(t))|^2.$

This mechanism yields equidistant, phase-coherent spectral lines, facilitating advanced frequency comb applications, pulse recompression, and next-generation telecommunications and spectroscopy.

6. Stokes Soliton Structures in Finite-Depth Fluids

Analysis of traveling wave solutions of the Euler equations for finite-depth fluids via the Babenko equation reveals that in the shallow water limit, Stokes waves transition to profiles akin to Korteweg–de Vries (KdV) solitons or cnoidal waves, but differ in their extreme form (Semenova et al., 28 Nov 2024). Numerical studies using Newton-Conjugate-Gradient and Fourier-Floquet-Hill techniques confirm:

As wave steepness increases, regardless of depth, Stokes waves develop a crest with a $2\pi/3$ angle, which is a geometric singularity absent in classical (smooth) KdV solitons.
In shallow water, wave profiles fit soliton-like or cnoidal wave functions over most of their support, but the sharp crest feature distinguishes Stokes-type solitons at high steepness. These findings connect solitary wave theory in shallow water with the structure and geometric singularities of classic Stokes waves.

7. Soliton Solutions in Viscid Incompressible Flow and Algebraic Structures

Transformation of the incompressible, viscid Navier–Stokes equations via repeated curl operations reduces them to higher-order diffusion equations, eventually bringing them into correspondence with the stationary KdV–Burgers equation (Meulens, 30 Jun 2025). The Weierstrass $\wp$ -function naturally emerges as a soliton solution: $\wp'(z)^2 = 4\wp^3(z) - g_2\wp(z) - g_3,$ mirroring the algebraic integrability of the stationary KdV. Higher-order derivatives yield N-soliton solutions analogous to those of the Kadomtsev–Petviashvili (KP) equation. The algebraic structure and interlockings of vorticity derivatives produce energetically-bounded, smooth, incompressible flow fields, suggesting deep interplay between coherent soliton structures and the integrability of Navier–Stokes-derived soliton equations.

In sum, the Stokes soliton concept unifies several advanced mathematical and physical constructs: it captures the essential role of beyond-all-orders asymptotics, spectral background modes, and nonlinear resonance in forming persistent, albeit subtly structured, localized wave states in hydrodynamic, optical, and statistical spectral systems. Theoretical and numerical evidence across these domains establishes the Stokes soliton as a robust, structurally intricate solitary object, whose existence and properties are intimately tied to the activation and smooth switching-on of exponentially small contributions—embodying the Stokes phenomenon across diverse nonlinear wave phenomena.