Orbital Stability of Solitary Waves

• Orbital stability of solitary waves is defined as the property where small, localized perturbations result in solutions that remain close to the solitary profile modulo symmetry actions.
• Analytical and numerical techniques leveraging variational characterizations and spectral analysis confirm a one-negative eigenvalue and coercivity conditions vital for stability proofs.
• Stability insights extend to nonlinear dispersive models such as NLS, Camassa–Holm, and Boussinesq systems, influencing research on asymptotic and structural stability.

Orbital stability of solitary waves is a fundamental concept in mathematical physics and nonlinear PDE theory, capturing the persistence of coherent structures in nonlinear dispersive evolution equations under small perturbations. A solitary wave is said to be orbitally stable if all sufficiently small, localized perturbations of the solitary profile yield global-in-time solutions that remain close—modulo symmetries such as translation and phase rotation—to the solitary-wave manifold under the induced group action. The rigorous analysis of orbital stability combines variational characterizations, spectral theory of linearized operators, refined functional inequalities, and dynamical Lyapunov functionals. The concept is central to the understanding of nonlinear dispersive models ranging from the nonlinear Schrödinger (NLS), Korteweg–de Vries (KdV), and Boussinesq systems to integrable and non-integrable shallow water equations and nonlocal models.

1. Variational and Spectral Foundations of Orbital Stability

Orbital stability in the energy space is typically established via the Grillakis–Shatah–Strauss (GSS) theory or its extensions. The foundational approach involves:

• Variational Characterization: Solitary waves are identified as (constrained) critical points of an action functional or Lyapunov-type functional, often derived from Hamiltonian and conservation laws of the respective PDE. For example, energy minimization under mass and/or kinetic constraints is standard in NLS-type models (Jin et al., 2021).
• Spectral Analysis of the Linearized Operator: Orbital stability requires the second variation (Hessian) of the action around the solitary wave to have precisely one simple negative eigenvalue, a simple kernel due to symmetries, and the remaining spectrum positive and bounded away from zero. Explicit identification of this spectral configuration is critical (Ma et al., 26 Dec 2025, Li et al., 2020, Guo, 2017, Long et al., 2022, Kwon et al., 2016).
• Coercivity with Constraints: Coercivity of the second variation on the orthogonal complement of the neutral directions must be established, often with respect to constraints reflecting the conservation laws (e.g., mass, momentum, or generalized Casimirs).
• Slope Conditions (Vakhitov–Kolokolov or Generalizations): A monotonicity or sign condition on a suitable "mass-frequency," "momentum-speed," or more general constraint curve is critical. Canonically, stability is linked to the sign of $\frac{d}{d\lambda} Q(\phi_\lambda)$ or, in multi-parameter settings, the signature of an associated Hessian matrix (e.g., $d''(\omega, c)$ in gDNLS (Liu et al., 2012), $\frac{d}{d k} Q(\phi)$ in $b$-CH (Long et al., 2022, Lafortune et al., 2022, Li et al., 2023)).

2. Methodologies Across Model Classes

The rigorous demonstration of orbital stability displays rich methodological diversity across models:

• Nonlinear Schrödinger-type Models: For mass-critical and supercritical pseudo-relativistic NLS in one dimension, Hong & Jin establish orbital stability via a constrained minimization (mass and kinetic energy), an improved Gagliardo–Nirenberg inequality uniform in the relativistic parameter, and a precise non-relativistic limit analysis. Local uniqueness and spectral coercivity are deduced by perturbation from the limiting nonrelativistic NLS ground state, leveraging variational compactness and the sharp structure of the linearized operator. The stability proof closes by a Lyapunov argument for the augmented energy, exploiting mass and energy conservation and spectral gap properties (Jin et al., 2021).
• Camassa–Holm and Generalized Equations: For the $b$-family of Camassa–Holm equations, the approach is to recast the problem as a planar Hamiltonian system, analyze the period function's monotonicity, and translate this into a derivative criterion on a period-like integral. The GSS/post-Weinstein orbital stability framework is adapted to the non-integrable context by showing that the monotonicity condition (e.g., $dQ/dk > 0$) holds analytically for all $b > 1$, thus settling longstanding open problems about non-integrable orbital stability (Long et al., 2022, Lafortune et al., 2022, Li et al., 2023).
• Systems and Nonintegrable PDEs: For coupled or nonintegrable systems such as the Degasperis–Procesi equation, the Schrödinger–Boussinesq system, or Good Boussinesq equations, the Lyapunov functional approach is enabled by identifying a variational structure, ensuring control over higher-order nonlinear terms via conservation laws, and deploying advanced a priori estimates that relate orbital norms (e.g., $L^\infty$ bounds in terms of $L^2$ norms). Spectral analysis demonstrates the requisite one-negative-eigenvalue/one-kernel configuration of the linearized Hessian, with coercivity on constraint-manifolds linked to the non-degeneracy of certain energy-momentum maps (Li et al., 2020, Ma et al., 26 Dec 2025, Maulén et al., 23 Jan 2025).
• Fourth-Order and Generalized Schrödinger Equations: Stability for fourth-order NLS models with mixed dispersion involves numerical and variational construction of smooth soliton branches, computation of VK-type derivatives, and spectral analysis of mixed-order operators. The critical exponent for stability may not coincide with the energy-critical power due to higher-order dispersion, requiring careful numerical continuity analysis and spectral diagnostics (Borluk et al., 2024).

3. Key Analytical Elements and Inequalities

A consistent feature in orbital-stability proofs is the deployment of sharp functional and interpolation inequalities:

• Improved Gagliardo–Nirenberg Inequalities: Uniform-in-parameter bounds, for example, those involving pseudo-relativistic kinetic energy, are necessary for uniform control of the nonlinearity in energy-critical and supercritical regimes (Jin et al., 2021).
• Conservative Structures and Casimirs: Multiple Casimirs (in addition to the Hamiltonian) often appear in noncanonical models (e.g., modified Camassa–Holm with cubic nonlinearity (Deng et al., 2024), Novikov equation (Ehrman et al., 2024)), requiring the construction of augmented Lyapunov functionals that incorporate the needed constraints for variational characterization.
• Period Function Monotonicity for Planar Systems: For certain hydrodynamical models, orbital stability reduces to the sign of a scalar integral associated with the period function of the reduced Hamiltonian system. The verification involves intricate algebraic and asymptotic analysis, including sign arguments on polynomial inequalities and numerical shooting (Long et al., 2022, Li et al., 2023).

4. Spectral Signature and Modulation Arguments

The spectral characterization of the linearized operator is indispensable:

• One Negative Eigenvalue, Simple Kernel, Positivity Elsewhere: This configuration is necessary for stability and is established either through direct Sturm–Liouville analysis, variational min–max principles, or numerical Evans function evaluation (the latter particularly for nonexplicit or nonlocal spectral problems as in the Novikov or pseudo-relativistic NLS equations (Ehrman et al., 2024, Jin et al., 2021)).
• Group-Orbit Structure: Modulation theory is utilized to decompose perturbations into coordinates along the group manifold (translation, phase, scaling) and transverse directions, ensuring invariance under the symmetries and orthogonality to kernel modes in the spectral analysis (Guo, 2017, Kwon et al., 2016, Maulén et al., 23 Jan 2025).

5. Model-Specific Applications and Extensions

Orbital stability has been rigorously established for a range of nonlinear dispersive systems, with representative results including:

Equation/Class	Stability Mechanism	Key Criterion/Analysis
Pseudo-relativistic NLS (1D)	Variational minimizer under constraints, spectral	Existence via GN, nonrelativistic limit, coercivity
Camassa–Holm b-family ($b > 1$)	Period monotonicity of planar ODE	$dQ/dk > 0$ via monotonicity (analytic+asymptotic)
Degasperis–Procesi	Lyapunov (augmented energy), $L^\infty$ control	Spectral theory, $L^\infty$–$L^2$ a priori estimate
Novikov, modified Camassa–Holm	Nonlocal action, explicit constraints (VK)	Spectral numerics, analytic VK sign condition
Good Boussinesq	Vector Lyapunov, virial, spectral decomposition	Matrix spectral signature, mixed-variable virials
Schrödinger–Boussinesq	Action functional, GSS framework	Index count, explicit profile integration
Fourth-order NLS	Numerical soliton curve, spectral VK criterion	$dQ/d\omega > 0$ threshold via numerical computation

These mechanisms have been extended to non-Hamiltonian systems (e.g., the Lugiato–Lefever equation (Bengel, 2023) via semigroup decay and contraction arguments), high-dimensional systems with nontrivial background or singular structures, and equations with nonlocalities and state-dependent Poisson maps (e.g., point-vortex water waves (Varholm et al., 2018, Gui et al., 10 Nov 2025)).

6. Resolution of Open Problems and Structural Stability

One significant outcome in recent literature has been the complete analytic resolution of the orbital stability question for all smooth solitary waves in the non-integrable $b$-family of Camassa–Holm models ($b>1$), previously open due to the lack of variational structure and explicit formulas beyond $b=2,3$. The method of recasting the orbital stability criterion in terms of period-like integrals and verifying their monotonicity by a mix of analytic and computational means closes this longstanding problem (Long et al., 2022). Moreover, the structural stability of the orbital stability property under variation of model parameters (e.g., $b$ in the b-CH family) has also been established, showing the robustness of this qualitative behavior (Li et al., 2023).

7. Dynamical Consequences and Asymptotic Stability

Beyond Lyapunov (orbital) stability, finer dynamical properties such as asymptotic stability and decay of perturbations to the solitary manifold are accessible through advanced spectral gap and virial-type arguments. For instance, in the Lugiato–Lefever equation, the combination of semigroup resolvent estimates and Lyapunov–Schmidt reduction yields not only orbital but exponential convergence in the energy norm for all localized perturbations (Bengel, 2023). In the Good Boussinesq case, new virial identities adapted to the vector structure yield full asymptotic stability results for speeds beyond the critical threshold (Maulén et al., 23 Jan 2025).

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References:

• (Jin et al., 2021): Hong, Jin, "Orbital stability for the mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation"
• (Long et al., 2022, Lafortune et al., 2022, Li et al., 2023): Long, Liu; Lafortune, Pelinovsky, "Orbital stability of smooth solitary waves for the $b$-family of Camassa-Holm equations"
• (Bengel, 2023): Pogan et al., "Stability of solitary wave solutions in the Lugiato-Lefever equation"
• (Li et al., 2020): Liu, Wang; "Orbital Stability of Smooth Solitary Waves for the Degasperis-Procesi Equation"
• (Ma et al., 26 Dec 2025): Ma, Xiao, "Orbital stability of solitary waves for the Schrödinger-Boussinesq system"
• (Maulén et al., 23 Jan 2025): Kowalczyk, Muñoz, "On asymptotic stability of stable Good Boussinesq solitary waves"
• (Borluk et al., 2024): Borluk, Muslu, Natali, Pastor, "On the orbital stability of solitary waves for the fourth order nonlinear Schrödinger equation"
• (Kwon et al., 2016, Guo, 2017, Liu et al., 2012): Fukaya, Guo, Liu-Simpson-Sulem, "Orbital stability of solitary waves for derivative nonlinear Schrödinger equation(s)"
• (Ehrman et al., 2024): Johnson, Stewart, "Orbital Stability of Smooth Solitary Waves for the Novikov Equation"
• (Deng et al., 2024): Deng, Lafortune, Liu, "Orbital stability of smooth solitary waves for the modified Camassa-Holm equation"
• (Varholm et al., 2018, Gui et al., 10 Nov 2025): Shatah, Walsh, Zeng; Wu, Liu, et al., "On the stability of solitary water waves with a point vortex", "Stability of Solitary Capillary-Gravity Water Waves in Three Dimensions".

This cumulative body of work delineates the technical criteria and mechanisms ensuring the orbital stability of solitary waves across major integrable, near-integrable, and non-integrable nonlinear dispersive models.