Spectral analysis of small amplitude periodic $b$-Novikov equation under transverse perturbations

Abstract: This paper is devoted to the transverse stability problem for small-amplitude periodic traveling waves of the $b$-Novikov equation, which arises as a two-dimensional extension of the $b$-family hierarchy. The perturbations under consideration fall into two groups. One group matches the fundamental period of the underlying wave along the longitudinal direction, while the other consists of profiles that are bounded or localized in the transverse direction. We perform spectral analysis of the associated linearized operator and derive precise stability and instability thresholds. The resulting conditions exhibit intricate dependence on two key parameters including the cubic coupling strength $b$ and the wavenumber $k$.

• The paper establishes a Lyapunov–Schmidt reduction to construct even, 2π-periodic small-amplitude solutions with explicit expansion coefficients.
• It employs Fourier analysis to compute eigenvalues and Krein signatures for coplanar transverse perturbations, outlining precise instability criteria.
• The investigation extends to non-periodic perturbations using Floquet theory, unifying instability regimes based on the parameters b and k.

Spectral Stability of the Small-Amplitude Periodic $b$-Novikov Equation Under Transverse Perturbations

Introduction and Motivation

The study addresses the spectral stability of small-amplitude periodic traveling waves in the $b$-Novikov equation, a cubically nonlinear integrable PDE generalizing the Novikov equation. The work is motivated by the broader problem of transverse instability in multidimensional extensions of dispersive equations, a foundational question in the analysis of wave phenomena where purely one-dimensional analysis is insufficient. Previous literature establishes sophisticated spectral stability criteria for the KP-type and Camassa–Holm (CH) equations, but the $b$-Novikov equation—especially under two-dimensional perturbations—remains largely under-explored. The authors fill this gap, systematically quantifying how the cubic nonlinearity ($b$) and longitudinal wavenumber ($k$) impact the stability landscape for periodic wave trains under both coplanar (periodic) and non-periodic (localized or bounded) transverse disturbances.

Existence of Small-Amplitude Periodic Traveling Wave Solutions

The $b$-Novikov equation is extended to a $(2+1)$-dimensional form to accommodate transverse analysis: $$\left[ u_t-u_{xxt} - buu_xu_{xx} + (b+1)u^2u_x - u^2u_{xxx} \right]_x - u_{yy} = 0,$$ where $b$ is the cubic coupling parameter and $u$ is the wave profile depending on $b$0, $b$1, and $b$2.

The existence of small-amplitude periodic traveling wave solutions is established via a Lyapunov–Schmidt reduction, leveraging bifurcation theory. With appropriate scaling, even, $b$3-periodic solutions are constructed, and perturbative expansions up to $b$4 in the amplitude parameter $b$5 are derived. Explicit expressions are provided for the wave profile and speed in terms of $b$6, $b$7, and the amplitude, using analytic expressions for expansion coefficients. The constraint $b$8 is imposed throughout, consistent with the integrability and physical admissibility of the equation.

Spectral Problem Formulation

The spectral analysis focuses on linearizing the two-dimensional $b$9-Novikov equation at the periodic background. The resulting eigenvalue problem is reduced, depending on the perturbation type, to the analysis of the operator

$b$0

where $b$1 is the transverse wavenumber, $b$2 encodes the linearization around the periodic wave, and $b$3 projects onto mean-zero functions. The analysis distinguishes between three classes of perturbations: periodic (coplanar), localized, and bounded.

Periodic (Coplanar) Transverse Perturbations

For transverse perturbations with the same $b$4-periodicity as the background wave, the spectrum of the linearized operator is explicitly computed via Fourier expansion. The eigenvalues $b$5 and associated Krein signatures are found: $b$6 For $b$7 and $b$8, the Krein signatures are strictly positive. Spectral stability is then governed entirely by the small-$b$9 (long transverse wavelength) regime, where collisions at the origin can occur.

A detailed block-diagonalization and perturbative expansion near $b$0—the only modes where instability may arise—reveals that the critical threshold for instability is governed by the sign of

$b$1

For $b$2, $b$3. For $b$4, instability is possible only when $b$5 is below a critical curve $b$6, reflecting a complex dependence of the instability threshold on both $b$7 and $b$8. When instability occurs, the spectrum bifurcates off the imaginary axis, producing a pair of real eigenvalues of opposite sign for $b$9.

Non-Periodic Transverse Perturbations

For disturbances that are either localized or bounded in the transverse direction, the spectral analysis utilizes Floquet theory, reducing the problem to a family of Bloch eigenvalue problems parameterized by the Floquet exponent $k$0. Explicit expressions are found for the unperturbed operator's spectrum and Krein signatures for each Floquet mode. Instabilities can only arise from collisions involving the lowest modes ($k$1), and such collisions—and hence instabilities—are only possible in a bounded region of $k$2-space.

A refined matrix perturbation argument near collision points yields an explicit instability window, the width of which depends on $k$3, $k$4, and the amplitude $k$5. The instability condition is governed by the sign of $k$6 where

$k$7

$k$8

with $k$9 signaling the possibility of real eigenvalues (instability). A sharp, uniform criterion is established for when the system is stable to all non-periodic transverse perturbations, depending only on $b$0, $b$1, and their relation to explicit curves.

Unified Thresholds and Regimes

A key cumulative result is the precise unification of transverse stability regimes:

$b$2	$b$3 range for $b$4 (instability window)	All $b$5
$b$6	all $b$7	Yes
$b$8	$b$9	Yes

For $(2+1)$0, small-amplitude waves are always subject to a finite band of unstable transverse modes for all wavenumbers. For $(2+1)$1, the onset or disappearance of instability is controlled by $(2+1)$2. There are no parameter regimes in which $(2+1)$3 everywhere, i.e., where non-periodic perturbations are always stable, emphasizing the inescapability of transverse instability in the $(2+1)$4-Novikov system for any significant region in $(2+1)$5-space.

Implications and Future Developments

The identification of universal and parameter-dependent instability thresholds in the two-dimensional $(2+1)$6-Novikov context advances the theoretical paradigm for transverse instability in non-KP, non-CH, cubically nonlinear dispersive systems. The results provide explicit quantitative criteria suitable for both mathematical insight and the numerical exploration of nonlinear stability and pattern selection in shallow water and related continuum models. The analytical framework extends the toolset available for the spectral analysis of higher-order and multi-parameter dispersive PDEs and lays a foundation for nonlinear and orbital stability studies.

Future developments could include the extension to solitary wave and large-amplitude periodic solutions, nonlinear orbital stability, and the investigation of corresponding instability mechanisms in the full nonlinear regime, possibly via rigorous modulation equations or inverse scattering methods. The intricate dependence of instability on the interplay of $(2+1)$7 and $(2+1)$8 also points toward potential applications in the design of systems where transverse instability is an exploitable or avoidable property.

Conclusion

This work delivers a comprehensive spectral analysis of small-amplitude periodic traveling waves in the $(2+1)$9-Novikov equation under both periodic and non-periodic transverse perturbations. Through explicit computation, perturbation theory, and careful consideration of the operator spectrum, the authors detail the parametric dependence of transverse instabilities, delivering full characterization of stability boundaries in terms of $$\left[ u_t-u_{xxt} - buu_xu_{xx} + (b+1)u^2u_x - u^2u_{xxx} \right]_x - u_{yy} = 0,$$0, $$\left[ u_t-u_{xxt} - buu_xu_{xx} + (b+1)u^2u_x - u^2u_{xxx} \right]_x - u_{yy} = 0,$$1, and amplitude. These results both generalize and sharpen known instabilities for integrable dispersive PDEs beyond the KP and CH classes, enabling further exploration of nonlinear wave propagation in multidimensional settings.

---

Reference:

"Spectral analysis of small amplitude periodic $$\left[ u_t-u_{xxt} - buu_xu_{xx} + (b+1)u^2u_x - u^2u_{xxx} \right]_x - u_{yy} = 0,$$2-Novikov equation under transverse perturbations" (2607.03846)