Cusp-like Soliton Structures in Nonlinear Systems

Updated 24 September 2025

Cusp-like soliton structures are nonlinear wave excitations characterized by sharp cusps or discontinuities in their derivatives, arising from singular modulation and bifurcation phenomena.
Analytical and numerical methods, including variational approximations, parametric transforms, and weak solution frameworks, are employed to derive explicit profiles and assess stability.
Their multidisciplinary applications span optics, plasma physics, and cosmology, offering insights into energy localization, symmetry breaking, and topological control.

Cusp-like soliton structures are nonlinear wave excitations characterized by sharp or singular features, such as discontinuities in the first derivative (a true cusp) or an infinite slope at a localized point. They are observed across a broad spectrum of nonlinear systems, both integrable and non-integrable, and are associated with mechanisms including singular nonlinearity modulation, bifurcation phenomena, nonlocal interactions, and topological constraints. The prominence and persistence of cusp-like solitons have multidisciplinary significance in optics, condensed matter, plasma physics, fluid dynamics, and cosmological modeling.

1. Mathematical Genesis and Archetypes of Cusp-like Solitons

Cusp-like soliton profiles emerge when system nonlinearities, symmetry-breaking instabilities, or modulation constraints engender sharply localized energy or intensity profiles. Canonical settings include:

Singular Modulation of Nonlinearity: When the nonlinear coefficient in NLSEs or related models is modulated by a singular or cusp-shaped function (e.g., $|x|^{-\alpha}$ ), the solutions develop profiles with divergent or discontinuous derivatives at the spatial singularity. In models with a cubic Kerr nonlinearity modulated as $g(x) = |x|^{-\alpha}$ ( $0 < \alpha < 1$ ), this engenders “quasi-cuspon” solitons—a term signifying infinite second derivative and a finite but non-smooth first derivative at the center (Borovkova et al., 2012, Lutsky et al., 2017, Lutsky et al., 2015).
Bifurcation-induced “Cusp Loops”: In dual-core or lattice-coupled systems, the soliton branching structure forms closed loops in the bifurcation diagram (norm $E$ vs. asymmetry $\Theta$ ), with regions of double bistability and abrupt transitions analogous to “cusp catastrophes” (Dror et al., 2010, Petrovic et al., 2011). These “loops” represent sharp, cusp-like transitions between symmetric and asymmetric soliton families driven by symmetry-breaking pitchfork bifurcations.
Nonlocal and Topological Nonlinearities: In models where the nonlinear response depends on curvature or topological features (e.g., the number and persistence of local extrema), cusp-like soliton formation is energetically favored at locations of abrupt profile change. An explicit realization uses a nonlinearity $N[|\psi|^2] = \alpha \partial_x \operatorname{sgn}(\partial_x |\psi|^2)$ , where the response is discontinuous at the extrema, promoting robust cusp-like solitons (Cao et al., 23 Sep 2025).
Nonclassical Solitary Waves (Cuspons and Peakons): In modified Korteweg–de Vries equations supplemented by dissipative or nonlocal dispersive terms, “cuspons” appear as continuous waves with derivative singularities, in contrast to “peakons” (with bounded but discontinuous derivatives) and classical smooth solitons (Rodriguez et al., 2023, Matsuno, 2019).
Integrable Models with Cusp Solutions: Certain integrable systems, including generalized short pulse and long–short wave equations, admit exact “cusp soliton” solutions in closed parametric form. These are primary examples of soliton solutions where the wave amplitude remains finite but the spatial derivative diverges at an isolated point (Matsuno, 2019, Zhu et al., 2014).

2. Bifurcations, Symmetry Breaking, and Multistability

Cusp-like phenomena are closely linked to bifurcation theory and the structure of the underlying dynamical systems:

Bifurcation Loops and Cusp Catastrophes

Dual-Core/Coupled Systems: Symmetry-breaking pitchfork bifurcations manifest as abrupt transitions from symmetric to asymmetric soliton states. Closed bifurcation loops in $(E,\Theta)$ or $(\epsilon,A,B)$ parameter spaces often display “cusp”-like profiles, particularly when direct and reverse bifurcations are both subcritical. These are observed for both fundamental and vortex soliton families (Dror et al., 2010, Petrovic et al., 2011).
Parameter Space Folding: Solutions exhibit folding or coexistence of multiple stable branches—characteristic of the cusp singularity and universal unfolding, as seen in parameter regions with bistability or double bistability (“double folding” in the loop) (Dror et al., 2010, Parra-Rivas et al., 2022).

Symmetry Breaking in Singular and Lattice Systems

Spontaneous Symmetry Breaking (SSB): In systems with symmetric singular modulation (e.g., two or more symmetric $|x \pm \Delta|^{-\alpha}$ peaks), increasing the propagation constant (or nonlinearity strength) induces SSB from symmetric to asymmetric localized modes via a supercritical pitchfork bifurcation. The bifurcation is often accompanied by formation of cusp-like features in the solution profiles and abrupt transitions in asymmetry parameters (Lutsky et al., 2017, Lutsky et al., 2015, Petrovic et al., 2011).
Pitchfork Bifurcation in Discrete Lattices: In interface-coupled two-dimensional lattices, symmetric soliton complexes lose stability at a critical coupling and transition sharply to stable asymmetric solutions, with bistability regions evident and dynamics reminiscent of cusp catastrophes (Petrovic et al., 2011).

3. Classification by Regularity and Singularity

A rigorous taxonomy distinguishes the regularity features of various solitary wave structures:

Type	Continuity	Derivative at Peak	Origin/Emergence
Classical Soliton	$C^\infty$	Finite, smooth	Standard balance of nonlinearity and dispersion
Peakon	Continuous	Discontinuous but bounded	Nonlinear dispersion (e.g., Camassa–Holm)
Cuspon	Continuous	Divergent (infinite slope)	Nonlinear dispersion and singular response
Quasi-cuspon	Continuous	Finite, $w''$ diverges	Singular modulation in NLSE

The formation of nonclassical solitons (cuspons/peakons) requires models with suitable weak (distributional) solution frameworks, where the jump conditions at the crest of the soliton are dictated by the model parameters and nonlocal/dissipative terms (Rodriguez et al., 2023, Zhu et al., 2014, Matsuno, 2019).

4. Stability, Instabilities, and Dynamical Evolution

Linear Stability Analysis: In dual-core and lattice systems, stability is systematically determined via perturbation theory, with eigenvalue analysis of small fluctuations (including azimuthal indices for vortex rings). For singularly modulated NLSEs, the Vakhitov–Kolokolov criterion and full spectral analysis together delineate stable and unstable regions for quasi-cuspon and cusp-like modes (Dror et al., 2010, Lutsky et al., 2017, Borovkova et al., 2012, Lutsky et al., 2015).
Instability Dynamics: Instabilities in vortex soliton rings (especially with high azimuthal perturbation indices) lead to azimuthal splitting, with the number of fragments determined by the dominant perturbation mode. Symmetry-breaking instabilities in interface and dual-core systems generate oscillatory or breathing asymmetric states (Dror et al., 2010, Petrovic et al., 2011).
Interaction and Scattering: Cusp-like soliton structures induced by singular nonlinear modulation demonstrate robust interaction characteristics, including resonance (capture by the singular peak if propagation constant matches), reflection, and destruction at critical velocities. Collision-induced dynamics comprise reflection, inelastic merging, and, in some cases, formation of persistent breathers or asymmetric pinned states (Lutsky et al., 2017).

5. Analytical and Numerical Construction

Variational Approximations: Variational methods employing suitable, often non-smooth, ansätze (e.g., Gaussians, exponential decay, or $|x|^{-\alpha}$ for singular modulation) are applied to derive amplitude, width, and critical parameter relations, as well as thresholds for existence and bifurcation points (Dror et al., 2010, Petrovic et al., 2011, Borovkova et al., 2012, Lutsky et al., 2015).
Parametric and Reciprocal Transformations: For generalized short pulse and long–short wave equations, hodograph and tau-function methods yield explicit parametric forms for cusp solitons, with the divergence of spatial derivative manifest at critical “trough” or “crest” positions (Matsuno, 2019, Zhu et al., 2014).
Weak Solution and Distributional Approaches: For generalized mKdV-type equations with dissipation and nonlinear dispersion, the profile is constructed in the space of distributions, permitting singularities at the soliton peak (cuspons), with the singularity controlled by parameter $r$ in $u_x \sim |x|^{r-1}$ (Rodriguez et al., 2023). Consistency is assured by imposing algebraic (jump) conditions at the singular point to ensure the solution remains weakly valid.

6. Physical Realizations and Applications

Cusp-like soliton structures appear in a range of physical contexts:

Nonlinear Optics: Structured optical waveguides or photonic lattices, especially with engineered Kerr or quadratic nonlinearity profiles, can stabilize cusp-like and flat-top solitons, enabling power-controlled optical switching, robust beam shaping, and energy localization beyond standard modulational instability thresholds (Lutsky et al., 2017, Borovkova et al., 2012, Lutsky et al., 2015, Matsuno, 2019, Cao et al., 23 Sep 2025).
Bose–Einstein Condensates: Feshbach-resonance techniques in BECs allow spatial engineering of nonlinearities (density-dependent gauge fields), facilitating experiments with dual-core, pancake traps, or systems with singular/nonlocal nonlinear response. This supports the realization of localized matter-wave excitations with sharp or topologically robust features (Dror et al., 2010, Driben et al., 2014, Cao et al., 23 Sep 2025).
Plasma Physics: Laboratory plasma devices employing multi-pole cusp magnetic fields (MPD) have demonstrated ion acoustic solitons consistent with KdV theory; when nontrivial geometries or multicomponent plasmas are employed, additional control over soliton amplitude, width, and collision properties are possible, as evidenced by amplitude–width measurements and Mach number studies (Shaikh et al., 2022, Shaikh et al., 2023). Quantum plasma models (electron-positron-ion systems) offer theoretical frameworks for quantum cusp solitons via nonlocal Zakharov and NLS-type equations (Ehsan et al., 2017).
Cosmology and Astrophysics: In ultra-light axion dark matter models, ground-state soliton profiles with a flat (non-cusped) core offer a mechanism for solving the ‘cusp–core’ problem in galactic dark matter distribution, with soliton solutions representing a core region of the halo that differs from the standard NFW cusp (Marsh et al., 2015).
Hydrodynamics and Fluid Mechanics: The cycloidal limit of periodic cusp solitons in generalized short pulse equations is mathematically identical to the Gerstner trochoidal solution for deep water waves, establishing the relevance of these structures in modeling extreme nonlinear hydrodynamic phenomena (Matsuno, 2019).
Field Theory and Topological Models: In O(3)-sigma models with cuscuton-like contributions, topological soliton solutions acquire internal ring-like or cusp-like structures, dominated by non-canonical (impurity) terms in the energy density and influenced by non-minimal couplings to electromagnetic fields (Lima et al., 2023).

7. Topological and Persistent Homology Perspectives

Recent developments employ concepts from topological data analysis (persistent homology) to quantify the energetic and structural properties of solitons:

Topological Nonlinearity: By crafting the nonlinear response to depend on the persistence of features (local maximum/minimum differences), the energy of a solution becomes a sum over the lifetimes of critical points in the intensity profile. The soliton is thus “topologically charged,” with its stability and robustness deeply linked to a global topological descriptor (Cao et al., 23 Sep 2025).
Control and Stabilization: This approach enables the suppression of modulational instabilities for flat-top beams and the stabilization of sharp edges and cusp features, offering new paradigms for waveform engineering in both optics and cold atoms.

Cusp-like soliton structures thus constitute a broad and unifying theme in nonlinear science, encompassing abrupt symmetry-breaking bifurcations, singular modulation, nonlocal and topological nonlinearities, and connections across optics, condensed matter, plasma, and field theory. Their analytic description, experimental realization, and dynamical control remain subjects of intense and ongoing research.