Nonlinear Dispersive Wave Models

• Nonlinear dispersive wave models are partial differential equations that balance nonlinear advection with higher-order or nonlocal dispersion to capture solitary waves and shock formations.
• Advanced spectral methods and cosine-basis projections are employed to analyze bifurcation structures, stability thresholds, and singular solutions in models like the Whitham and Benjamin equations.
• These models elucidate phenomena such as cusp formation, stability inversion, and irreversible soliton interactions, offering insights into non-integrable dynamics in fluid, optical, and elastic systems.

Nonlinear dispersive wave models constitute a broad class of partial differential equations (PDEs) in which the nonlinear advection of wave fields is balanced by dispersive effects generated by nonlocal or higher-order differential operators. These models arise in fluid dynamics, elasticity, nonlinear optics, and granular media, and they exhibit phenomena such as solitary waves, dispersive shock waves, modulated wave trains, and singular solutions. Representative examples include the Whitham equation, the Benjamin–Ono equation, the Benjamin equation, and other nonlocal PDEs. The interplay between nonlinearity and dispersion leads to intricate bifurcation structures, stability inversions, and derivative singularities, which are best analyzed using advanced spectral and continuation techniques.

1. General Nonlinear Dispersive PDE Framework

The prototypical form for many nonlinear dispersive models is

$u_t + [f(u)]_x + \mathcal{L}u_x = 0,$

where $u(x,t)$ is the wave field, $f(u)$ a smooth nonlinearity with $f(0)=0$, and $\mathcal{L}$ a self-adjoint Fourier-multiplier operator, $$\mathcal{L}u = \mathcal{F}^{-1} [\alpha(k)\,\widehat u(k)]$$, with $\alpha(k)$ determining the dispersion relation. Model-specific choices include:

• Whitham equation: $\alpha(k) = \sqrt{\tanh(h_0 k)/k}$, $f(u) = \frac{3}{4}u^2$;
• Modified Benjamin–Ono: $\alpha(k)=1+|k|$, $f(u)=\frac{1}{3}u^3$;
• Benjamin equation: $\alpha(k)=1 - |k| + \tau k^2$, $f(u)=\frac{1}{2}u^2$.

Traveling-wave solutions $u(x,t)=\phi(x-ct)$ satisfy after one integration

$-c\,\phi + f(\phi) + \mathcal{L}\phi = B,$

with constant $B$ set by symmetry or mean constraints. Analysis of these profile equations underpins investigation of bifurcation diagrams, stability thresholds, and singular limiting profiles.

2. Spectral Continuation and Numerical Methodology (SpecTraVVave)

The steady profile $\phi(\xi)$ is projected onto a finite cosine basis: $$\phi_N(\xi) = \sum_{l=0}^{N-1}\Phi_l\cos(l\,\xi),\quad \xi\in[0,2\pi].$$ Discrete operators are encoded via diagonal Fourier-multiplier matrices $L^N_{ij}$, and collocation is enforced at $\xi_n = \pi(2n-1)/(2N)$. The nonlinear steady-state system

$$-c\,\phi_N(\xi_n) + f(\phi_N(\xi_n)) + (L^N\phi_N)(\xi_n) = B,$$

is augmented with amplitude and mean constraints, resulting in a system of dimension $N+2$.

A key feature is the secant-based continuation in $(c,a)$ parameter space, allowing traversal of folds and turning points. Given two reference points $P_1$, $P_2$ in the $(c,a)$ plane, a secant predictor is followed by Newton correction orthogonal to the secant direction, with residuals monitored in maximum norm and step-size adaptively controlled.

3. Case Studies: Representative Model Behaviors

3.1 Whitham Equation

The bifurcation branch in $(c,a)$ for $2\pi$-periodic waves is characterized by three distinguished points:

• Turning point: $(c_{min}, a_{turn}) \approx (0.798, 0.55)$;
• Stability inversion: $L^2$-norm maximum at $(c_{inv}, a_{inv}) \approx (0.812, 0.60)$, where $d''(c)$ changes sign;
• Cusped wave terminal point: $(c_{cusp}, a_{cusp}) \approx (0.835, 0.69)$, marked by $\phi_{max}/c \to 1.5$.

Cosine coefficients decay exponentially for smooth solutions, but switch to algebraic decay at the cusp, reflecting a derivative discontinuity.

3.2 Modified Benjamin–Ono Equation

Solitary-wave collisions reveal strong non-integrable behavior:

• Two pulses (speeds $c_1 > c_2$) collide; the smaller soliton is annihilated;
• An oscillatory dispersive tail is shed;
• Invariants $V$ and $E$ demonstrate net mass/energy transfer $\Delta V > 0$;
• Contrasts sharply with elastic soliton interactions in integrable models (cf. KdV).

3.3 Benjamin Equation

Here, two linear dispersive terms compete:

• Multiple bifurcation branches of periodic waves exist for distinct wavenumbers;
• Branches can cross and reconnect high up in the nonlinear regime, e.g., coincident profiles at $(c^*, a^*) \approx (0.945, 5.94)$ signal branch reconnection due to gravity–capillary balance ($\tau k^2$ vs. $|k|$).

4. Physical Phenomenology and Stability Criteria

The bifurcation and continuation analysis captures the influence of nonlinearity-dispersion balance:

• Weaker, nonlocal dispersion (as in Whitham) permits finite-amplitude cusp formation, while stronger, higher-order dispersion (as in KdV) enforces smoothness.
• Turning points in $(c,a)$ diagrams do not imply loss of stability; $d''(c)=0$ gives precise thresholds, with stability inversion at $L^2$-norm extrema.
• Formation of a cusp correlates with a transition from exponential to algebraic decay in modal amplitudes and the failure of regularity bootstrapping.

For solitary-wave dynamics:

• Non-integrable models with Hilbert transform dispersion and higher nonlinearities (e.g., cubic) can generate irreversible soliton destruction, tail radiation, and net energy transfer.

Bifurcation topology with competing dispersions gives rise to crossing/reconnection of solution branches, controlled by the relative strength and sign of dispersion terms.

5. Implementation Strategy and Model Extension

The SpecTraVVave approach generalizes to any equation of the form $u_t + [f(u)]_x + \mathcal{L}u_x = 0$ where $\mathcal{L}$ is a self-adjoint, nonlocal Fourier multiplier. A hierarchy of Fourier-multiplier kernels and nonlinearities is supported, enabling analysis of varied physical models.

Key ingredients for successful implementation:

• Correct choice of cosine basis and collocation grid for high spectral accuracy;
• Careful construction and differentiation of the discrete operator matrices for arbitrary $\alpha(k)$;
• Robust secant continuation and fold-traversal methods, with step-size and residual control;
• Stability verification via second derivative conditions (e.g., $d''(c)$) and analysis of $L^2$ norms.

This framework allows the systematic numerical exploration of traveling-wave branches, identification of global bifurcation structures, and quantitative investigation of nonlinear dispersive wave interactions, including both solitary and periodic regimes.

6. Significance and Broader Context

Nonlinear dispersive wave models such as those investigated with the SpecTraVVave code occupy a central position in modern PDE modeling of physical systems where nonlinearity and nonlocal dispersion are concurrently present. They offer insight into the mechanisms of stability inversion, singularity formation, and non-integrable interactions. The methods and findings summarized above are directly applicable to the analysis and simulation of nonlinear water waves, internal waves in stratified fluids, nonlinear optics, and granular crystal models.

Extensions of this framework provide a basis for incorporating bathymetric variability, variable coefficients, multi-dimensional generalizations, and coupling to external perturbations, enabling the paper of complex wave phenomena in realistic geophysical and engineering contexts.