Numerical Study on the Evolution of Peregrine Breathers in Variable Depths

Abstract

The Peregrine breather (PB), a classical localized solution of the nonlinear Schrödinger equation (NLSE), is widely used to describe the evolution of deep-water rogue waves. However, the influence of variable bathymetry on PB focusing remains insufficiently understood. A two-dimensional RANS–VOF numerical wave tank is therefore established using computational fluid dynamics (CFD) to investigate deterministic PB propagation over variable bathymetry. The model is validated through mesh- and time-step-sensitivity analyses and comparison with the analytical PB solution. Relative water depth, bathymetric interaction length, and bathymetric position are systematically examined. The results reveal for the first time a bathymetry-induced delayed-focusing phenomenon: the PB undergoes local defocusing over elevated topography and refocuses farther downstream after re-entering deeper water. The delay increases as water depth decreases. For k₀h_shelf > 1.363, increasing the interaction length mainly enhances the focusing delay, while self-focusing recovers in deeper water. In contrast, for k₀h_shelf < 1.363, an interaction length of approximately two carrier wavelengths disrupts the coherent PB structure and splits it into two wave packets. The onset position of bathymetric forcing has only a minor effect on the final delay. These results clarify how variable bathymetry modulates PB focusing and structural stability and provide a theoretical reference for nearshore extreme-wave risk assessment.

1. Introduction

Rogue waves are a critically important nonlinear physical phenomenon that has been extensively studied across diverse physical systems, including water waves, optical fibers, ultrasonic waves, and plasmas. In hydrodynamics, rogue waves are typically characterized as strongly nonlinear and highly asymmetric wave events. They are also referred to as “freak waves,” “extreme waves,” and related terms. Klinting et al. [1] defined rogue waves according to three criteria: the wave height exceeds twice the significant wave height; it is greater than twice the heights of the preceding and following waves; and the crest height exceeds 65% of the total wave height. However, satisfying all three criteria simultaneously in realistic ocean conditions is difficult. Kharif [2] proposed that under the assumption of a Gaussian sea state, a wave with a maximum wave height can be considered a rogue wave. The nonlinear behavior of rogue waves is associated with complex energy transfer and rapid waveform evolution, which cannot always be adequately represented by conventional linear wave theory. Another important characteristic is their strong transience: rogue waves may emerge rapidly from the surrounding wave field, reach an extreme amplitude over a short interval, and subsequently decay. This makes them a significant potential threat to marine engineering, ship navigation, and related fields [3]. Accordingly, considerable attention has been devoted to understanding the generation and evolution mechanisms of extreme waves [4,5,6].

Under a multiple-scale approximation, the weak nonlinear modulation of ocean carrier waves can be described by focusing-type nonlinear Schrödinger equations (NLSE). Building upon the work of Ma [7], Peregrine systematically investigated the relationship between water wave modulation, the NLSE, and its analytical solutions in 1983 [8]. Among these solutions is a class of rational-function analytical solutions with spatiotemporal localization properties, which was later named the Peregrine breather (PB). This solution exhibits strong nonlinear modulation: its amplitude can reach up to three times the background amplitude, and it shows the characteristic dynamics of abrupt appearance and disappearance [9]. It has thus become a crucial theoretical prototype for studying extreme wave events and has been suggested as a model for oceanic rogue waves [9]. Indeed, such waves “appear from nowhere and disappear without a trace,” much like rogues in the ocean [10]. Further research has revealed that the PB solution is closely related to modulation instability (MI) and can be viewed as a limiting case of MI, making it a key entry point for understanding rogue wave formation mechanisms [10,11]. Theoretically, the PB can be considered the limiting form of both Akhmediev breathers and Kuznetsov–Ma breathers [10]. Building on this foundation, subsequent research has gradually extended PB-related theory to conditions that more closely reflect realistic water-wave dynamics. For example, He and Gaillard [12,13] incorporated higher-order nonlinear effects and finite-depth corrections into generalized NLSE-type models, and explored the existence and evolution of Peregrine-type solutions in these models. Research by Cui et al. [14] demonstrates that higher-order Peregrine solutions based on the NLSE can reveal the nonlinear evolution and higher-harmonic characteristics of extreme water waves, offering new insights into discrepancies between theoretical predictions, numerical simulations, and experimental observations, as well as the formation mechanisms of extreme waves.

Water depth is also an important factor governing the evolution of PB. Under finite-depth conditions, variations in relative water depth modify the dispersive and nonlinear characteristics of the wave system, thereby affecting modulational instability (MI), crest amplification, and the self-focusing behavior of PB wave packets. Previous theoretical studies have shown that finite-depth corrections can significantly alter the evolution of Peregrine-type solutions [12,13]. In addition, Zeng and Trulsen [15] investigated weakly nonlinear, unidirectional wave propagation over a sloping bottom and showed that bathymetric variations can induce a dynamical response extending beyond the local region of depth change. They reported that narrow-banded wave trains are modulationally unstable when kh > 1.363, whereas long-crested waves become modulationally stable when kh < 1.363. Their results further showed that, after propagating over the slope, the wave field may require a finite downstream adjustment distance to adapt to the new water-depth environment, particularly when the shallower-water depth approaches or falls below the critical value kh ≈ 1.363.

These findings indicate that the influence of variable bathymetry is not necessarily confined to the local bathymetric transition but may persist farther downstream. When a deterministic PB propagates over variable bathymetry, the local wavenumber, group velocity, dispersion, and nonlinearity may vary along the propagation path as the water depth changes, thereby perturbing the dispersion–nonlinearity balance governing PB self-focusing. Consequently, the focusing position, crest amplification, and structural stability of the PB may all be modified. Nevertheless, although PB evolution under uniform deep- and finite-depth conditions has been relatively well studied, the propagation, focusing, and structural stability of deterministic PBs over variable bathymetry remain insufficiently understood.

Chabchoub et al. [16] first experimentally reproduced the evolution of a surface gravity wave consistent with the PB profile in a laboratory wave flume, providing important experimental support for subsequent studies of rogue-wave dynamics. Chabchoub et al., Onorato et al., and Zhang et al. [17,18,19,20] subsequently investigated PBs experimentally and examined their fundamental evolution characteristics. Perić et al. and Klein [21,22] generated Peregrine-type rogue waves in physical wave tanks and reported good agreement between the measured waveforms and theoretical predictions. Dong et al. [23] conducted a series of experiments in a finite-depth flume, investigating the evolution characteristics of PB under different k₀h conditions (where k₀ is the wavenumber and h is the water depth). Their experiments examined features including waveform evolution and propagation drift. Liao et al. [24] analyzed the effect of a uniform adverse current on PB evolution in flume experiments and found that opposing currents accelerate wave-packet focusing and lead to earlier rogue-wave formation. Zhang et al. [25] proposed a novel experimental method that successfully generated PB in deep-water conditions. Using free-surface measurements and wavelet analysis, they reported close agreement between the experimental observations and analytical solutions and provided further insight into modulational instability (MI) and wave-energy focusing.

Perić et al. [21] conducted direct numerical simulations of Peregrine-type waves using a two-phase Navier–Stokes solver based on the finite-volume method and the volume-of-fluid (VOF) approach. The simulation results showed excellent agreement with experiments, not only confirming the reliability of the numerical method in capturing MI and breaking processes but also revealing, for the first time, new characteristics in the underwater flow field. These features may provide new clues for the short-term prediction of rogue waves, and repeated wave breaking was observed. Hu et al. [26,27] successfully performed numerical simulations of rogue waves in finite depth and investigated their time-frequency characteristics. This study derived the second-order Stokes component wave expression for the PB and used it as a theoretical basis for numerical reconstruction. Qin et al. [28] constructed a two-dimensional (2D) numerical wave tank based on the PB solution for simulation, coupling the N-S equations with the finite element method. By comparing with regular waves, the study revealed the uniqueness of rogue wave impacts. Zhang et al. [29] used computational fluid dynamics (CFD) to numerically simulate PB waves, revealing the nonlinear relationship between impact pressure and wave-height growth; they also found that short-period rogue waves induce more severe impact effects than long-period waves. Through 294 numerical cases, Qin et al. [30] systematically analyzed the impact force of PB rogue waves on decks and proposed an empirical formulation for predicting maximum wave loads through parametric analysis. Tika et al. [31] introduced a “ nonlinear spectral engineering” strategy, using numerical simulations to design initial spectral distributions that enable the controllable generation of PB events in a water-wave field, thereby providing a pathway for experimental and engineering applications at the numerical level.

Non-uniform bathymetry is an important environmental factor governing wave evolution, and its influence on rogue-wave formation has attracted increasing attention in recent hydrodynamic research. Trulsen et al. [32] numerically studied wave evolution over a sloping seabed and showed how bathymetric variations can generate rogue waves at specific locations by altering the local Ursell number and the growth rate of Benjamin-Feir instability (BFI). Research by Trulsen et al. [33] showed that when long-crested irregular waves propagate from deep to shallow water, their dynamic characteristics undergo fundamental changes, leading to wave steepening and triggering nonlinear energy transfers that increase wave heights and may excite MI. Although previous studies have shown that relatively simple bathymetric variations can influence the occurrence of rogue waves [32], systematic observations of the complete interaction between a deterministic PB and specific bathymetric features, such as slopes or shoals, remain limited. Our previous work [34], through flume experiments, confirmed that stable, coherent localized wave packets can emerge within a random wave field based on a JONSWAP spectrum. Their characteristics align closely with various breather and soliton solutions of the NLSE, and their propagation over non-uniform bathymetry can be reasonably described within a variable-coefficient generalized NLSE framework.

The primary objective of this study is to investigate the influence of variable bathymetry on the evolution of PB waves and the characteristics of the resulting rogue waves. To this end, CFD simulations are performed in a two-dimensional numerical wave tank. The hydrodynamic processes are resolved by solving the incompressible Navier–Stokes equations coupled with the VOF method for free-surface reconstruction. This framework enables accurate simulation of water-wave dynamics and effectively captures nonlinear wave effects. In the present study, the PB solution of the NLSE is employed to generate nonlinear rogue waves, and their evolution under various conditions is systematically investigated through numerical simulations. Particular attention is given to the behavior of PB waves over spatially varying bathymetry, with emphasis on the effects of different water depths, different durations of bathymetric forcing, and different stages of PB evolution on wave dispersion and nonlinear spatial modulation, thereby revealing the regulating effects of uniform bathymetry on PB breaking, focusing, and dissipation. The principal novelty of this study lies in the first identification and systematic characterization of a bathymetry-induced delayed-focusing phenomenon: the coherent focusing process of the PB is locally suppressed over elevated bathymetry, whereas the focusing process can recover farther downstream after the wave packet re-enters deeper water. The numerical results further demonstrate that both the focusing delay and structural stability of the PB depend strongly on the relative water depth and bathymetric interaction length, and that sufficiently strong bathymetric forcing may disrupt the coherent PB structure and even induce wave-packet splitting. These findings provide a theoretical basis for the prediction of extreme waves in nearshore regions and for disaster prevention and mitigation associated with wave impacts.

2. Nonlinear Schrödinger Equation and Peregrine Breather Solutions

To describe the nonlinear dynamics of surface gravity waves, including rogue-wave evolution in numerical wave tanks, various theoretical and numerical approaches have been developed. Among them, the nonlinear Schrödinger equation (NLSE) is widely used to describe the evolution of slowly modulated, weakly nonlinear wave trains. The NLSE framework is applicable to both deep-water and finite-depth conditions, with its dispersion and nonlinear coefficients depending on water depth. The detailed derivation is given by Mei [35]; therefore, only the expressions required for the subsequent analysis are presented here. Within this weakly nonlinear perturbation framework, Equations (1) and (2) represent the first-order velocity potential ${\phi }_1$ and the first-order free-surface elevation ${\zeta }_1$, respectively. These expressions describe the leading-order wave kinematics and free-surface behavior of the modulated surface-gravity-wave field and serve as the basis for constructing the analytical free-surface elevation and velocity field imposed at the inlet of the numerical wave tank.

$${\phi }_1={\phi }_{10}-{\displaystyle \frac{g\mathit{cosh}Q}{2\omega \mathit{cosh}q}}(iA{e}^{i\psi }-i\overline{A}{e}^{-i\psi })$$

(1)

$${\zeta }_1={\displaystyle \frac{1}{2}}(A{e}^{i\psi }-i\overline{A}{e}^{-i\psi })$$

(2)

where ${\omega }^2=gk\mathit{tanh}(kh)$, Q = k(z + h), q = kh. It is important to note that the origin of the coordinate system is placed at the still water surface; thus, z = 0 denotes the mean water level, while z = −h denotes the bottom. The symbols h, g, k, $\omega$ and $\psi$ denote the water depth, gravitational acceleration, carrier wavenumber, angular frequency, and wave phase, respectively. The quantities ${\phi }_{10}$ and A denote the mean flow and the carrier-wave envelope, respectively. The latter is governed by the well-known NLSE [26,27]:

$$-i{\displaystyle \frac{\partial A}{\partial \tau }}+\alpha {\displaystyle \frac{{\partial }^{2}A}{\partial {\xi }^2}}+\beta {\left|A\right|}^{2}A=0$$

(3)

where $\xi =x_1-C_g{t}_1$, $\epsilon =k{A}_0$, $\tau ={\epsilon }^{2}t$, $C_g={\displaystyle \frac{\partial \omega }{\partial k}}$, C_g is the group velocity, A₀ is the parameter of the carrier amplitude.

Under the condition of infinite water depth, the expressions for parameters α and β are [11,27]

$$\alpha ={\displaystyle \frac{\omega }{8k^2}}$$

(4)

$$\beta ={\displaystyle \frac{\omega k^2}{2}}$$

(5)

Under finite water depth conditions, the expressions for parameters α and β are [27,35]

$$\alpha ={\displaystyle \frac{{C_g}^2}{2\omega }}-{\displaystyle \frac{\omega q\mathit{cosh}q}{k^2\mathit{sinh}2q}}+{\displaystyle \frac{q\mathit{sinh}q}{k\mathit{cosh}q}}{C}_g$$

(6)

$$\beta ={\displaystyle \frac{\omega k^2}{16{\mathit{sinh}}^{4}q}}(\mathit{cosh}4q+8-2{\mathit{tanh}}^{2}q)+{\displaystyle \frac{\omega }{2{\mathit{sinh}}^{2}2q}}{\displaystyle \frac{{(2\omega \mathit{cosh}2q+k{C}_g)}^2}{gh-C_g^2}}$$

(7)

It should be noted that Equations (5) and (7) represent the nonlinear coefficient β under different water-depth regimes. For deep-water waves, the cubic NLSE reduces to the conventional form with $\beta =\omega k^2/2$, which has been widely adopted in theoretical and experimental studies of PBs. In contrast, under finite-water-depth conditions, β depends explicitly on the relative water depth q = kh because finite-depth effects modify the nonlinear wave interactions, as expressed in Equation (7). In the infinite-depth limit, the finite-depth expression asymptotically approaches $\beta =\omega k^2/2$, thereby recovering the deep-water formulation. Therefore, Equations (5) and (7) are fully consistent: Equation (5) represents the deep-water limit, whereas Equation (7) gives the corresponding finite-depth form.

Peregrine [8] investigated a standard form of the NLSE, which is expressed as

$$i{\eta }_T+{\eta }_{XX}+2{\left|\eta \right|}^2\eta =0$$

(8)

By applying the following coordinate transformation, Equations (3) and (9) can be obtained:

$$T=-\tau , X={\alpha }^{-{\displaystyle \frac{1}{2}}\xi }, \eta ={\left({\displaystyle \frac{\beta }{2}}\right)}^{{\displaystyle \frac{1}{2}}}A$$

(9)

where T and X are the transformed temporal and spatial coordinates, respectively, and $\eta$ is the scaled complex wave envelope of the standard NLSE.

Based on the work of Ma [7], Peregrine first conducted a theoretical analysis of the amplitude peak behavior by means of a double Taylor series expansion. Subsequently, through an inverse coordinate transformation, he derived the breather solution for the standard NLSE, whose expression is given as follows:

$$A=A_0{e}^{-i\beta {A_0}^2\tau }[{\displaystyle \frac{4\alpha (1-2i\beta A_0^2\epsilon t_1)}{\alpha +\alpha {(2\beta A_0^2\epsilon t_1)}^2+2\beta A_0^2{\xi }^2}}-1]$$

(10)

First-order horizontal velocity of water particle u, vertical velocity of water particle w, and dynamic pressure p:

$$u={\displaystyle \frac{\partial \phi }{\partial x}}={\displaystyle \frac{gk\mathit{cosh}Q}{2\omega \mathit{cosh}q}}(A{e}^{i\psi }-i\overline{A}{e}^{-i\psi })$$

(11)

$$w={\displaystyle \frac{\partial \phi }{\partial z}}=-{\displaystyle \frac{gk\mathit{sinh}Q}{2\omega \mathit{cosh}q}}(iA{e}^{i\psi }-i\overline{A}{e}^{-i\psi })$$

(12)

$$p=-\rho gz+\rho g{\displaystyle \frac{\mathit{cosh}Q}{2\mathit{cosh}q}}(A{e}^{i\psi }-i\overline{A}{e}^{-i\psi })$$

(13)

More details regarding the derivation of the PB solution can be found in the relevant literature [8,17,26].

3. Numerical Simulation

3.1. Computational Mathematical Models

Over recent decades, with the continuous advancement of computer technology and numerical algorithms, CFD has undergone rapid development. Due to its high-precision numerical methods, well-developed physical models, and powerful post-processing capabilities, CFD has played a significant role in simulating the complex interactions between waves and coastal structures. Against this background, this paper is dedicated to simulating the PB with nonlinear characteristics in a numerical wave flume. It will systematically introduce the fundamental theories underlying this simulation, primarily including governing equations, free-surface capturing methods, as well as wave-generation and absorption methods.

(1)

Assuming the fluid is a two-dimensional, incompressible, viscous fluid, its fundamental governing equations are

$${\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial w}{\partial z}}=0$$

(14)

where u is the velocity component in the horizontal direction, and w is the velocity component in the vertical direction.

$${\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial z}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial x}}+u({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial z^2}})$$

(15)

$${\displaystyle \frac{\partial w}{\partial t}}+u{\displaystyle \frac{\partial w}{\partial x}}+v{\displaystyle \frac{\partial w}{\partial z}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial z}}+u({\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}w}{\partial z^2}})+g$$

(16)

where $\rho$ is the fluid density, t is time, u and w are the velocity components in the x and z directions, respectively, and $-{\displaystyle \frac{1}{\rho }}\nabla p$ is the pressure gradient.

(2)

Initial and boundary conditions for the numerical wave flume

The boundary conditions for the two-dimensional wave flume are as follows:

1.

Numerical flume bottom (Z = 0) satisfies the no-slip boundary condition:

$${\displaystyle \frac{\partial \phi }{\partial z}}{|}_{Z=0}=0$$

(17)

2.

For reference, in the analytical potential-flow formulation used to derive the incident PB solution, the free surface $\eta (x,t)$ satisfies the following kinematic and dynamic boundary conditions:

$${\displaystyle \frac{\partial \eta }{\partial t}}+{\displaystyle \frac{\partial \eta }{\partial x}}{\displaystyle \frac{\partial \phi }{\partial x}}{|}_{Z=\eta }+{\displaystyle \frac{\partial \eta }{\partial y}}{\displaystyle \frac{\partial \phi }{\partial y}}{|}_{Z=\eta }=0$$

(18)

$${\displaystyle \frac{\partial \phi }{\partial t}}{|}_{Z=\eta }+{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{\partial \phi }{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial \phi }{\partial y}}\right)}^2\right]{|}_{Z=\eta }+g\eta =0$$

(19)

where $\eta (x,t)$ represents the wave surface equation.

However, in the present CFD simulations, Equations (18) and (19) are not directly imposed as numerical free-surface boundary conditions. The numerical wave flume is solved using the Reynolds-averaged Navier–Stokes (RANS) equations coupled with the VOF method. The air–water interface is captured through the transport of the water volume fraction F; such that the interface motion is represented by the advection of the volume-fraction field by the local flow velocity. The interfacial dynamic response is resolved through the pressure and viscous-stress fields obtained from the two-phase RANS momentum equations, while the upper boundary of the computational domain is specified as a pressure outlet at atmospheric pressure. Thus, the potential-flow free-surface conditions are used only to construct the analytical incident-wave elevation and velocity field prescribed at the inlet, rather than being directly imposed as explicit free-surface boundary conditions in the RANS-VOF simulation.

3.

The boundary condition at the wave inlet in the x-direction of the tank is set as a velocity inlet; the boundary condition at the wave outlet in the x-direction is set as a pressure outlet; the two sidewalls in the y-direction are set as symmetric boundaries; the top surface in the z-direction is set as a pressure outlet boundary, with the pressure set to the standard atmospheric pressure.

The initial conditions of the two-dimensional wave tank are as follows:

The numerical wave tank starts from a state of rest, meaning the initial wave surface is the still water level, the initial velocity field is 0, and the fluid pressure is hydrostatic pressure.

(3)

Turbulence model

The $SSTk-\omega$ turbulence model proposed by Menter [36] was adopted in this study. Although turbulence effects are not equally significant throughout the wave field, the PB develops locally steep crests during the focusing process, with a baseline wave steepness of ${\epsilon }_0=a_0{k}_0=0.1258$. Meanwhile, interaction between the waves and variable bathymetry may significantly enhance local velocity gradients and shear, potentially accompanied by flow separation and vorticity generation. Therefore, compared with a purely laminar formulation, the introduction of turbulence closure provides a more reasonable representation of turbulence-related momentum transport and energy dissipation in these local regions. The $SSTk-\omega$ model has good near-wall performance and is suitable for flows involving adverse pressure gradients and local separation, making it appropriate for the bathymetry-interaction conditions considered in the present numerical wave tank. Compared with LES, this $SSTk-\omega$ model offers a more practical balance between computational cost and the representation of turbulence effects, and is therefore better suited to the present parametric study. It should be noted that the RANS turbulence closure may still affect the prediction of local energy dissipation; therefore, the results are mainly interpreted in terms of the relative variations among different cases. The transport equations for turbulent kinetic energy and specific dissipation rate are given as follows:

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho k_t\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho k_t{u}_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left({\Gamma }_k{\displaystyle \frac{\partial k_t}{\partial x_j}}\right)+G_k-Y_k+S_k$$

(20)

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho w_t\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho w_t{u}_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left({\Gamma }_w{\displaystyle \frac{\partial w_t}{\partial x_j}}\right)+G_w-Y_w+S_w$$

(21)

where k_t is the turbulent kinetic energy, w_t is the specific dissipation rate; G_k is the turbulent kinetic energy generated by the mean velocity gradient; G_w is the production term in the specific dissipation rate w_t equation; ${\Gamma }_k$ and ${\Gamma }_w$ are the diffusion coefficients for k_t and w_t; Y_k and Y_w represent the turbulent diffusion terms; S_k and S_w are user-defined source terms.

(4)

Free-surface capturing

In this paper, the VOF model is employed to capture the air–water interface. A volume fraction function, denoted as F, is introduced in each control volume of the computational domain to represent the local water-phase fraction. For two-phase flow, F = 1 indicates that the cell is completely occupied by water, F = 0 indicates that it is occupied by air, and 0 < F < 1 indicates that the gas-water interface crosses the cell. Therefore, the free surface is not prescribed explicitly as a potential-flow boundary condition in the numerical model; instead, it is captured and reconstructed from the evolution of the volume-fraction field within the RANS–VOF framework.

At the inlet of the numerical wave tank, a velocity-inlet boundary condition is employed for PB generation. The analytical PB velocity field and free-surface elevation are prescribed directly at the inlet, thereby avoiding errors associated with linear wavemaker transfer functions. Compared with moving-wavemaker approaches, such as piston- or flap-type wavemakers, this method does not require overset or deforming meshes, thereby facilitating mesh-quality control and reducing computational cost. It should be emphasized that the analytical PB solution provides only the incident-wave prescription at the inlet. After the wave enters the computational domain, the subsequent wave propagation and the evolution of the air–water interface are resolved by the RANS equations coupled with the VOF method rather than by potential-flow free-surface boundary conditions. Accordingly, the velocity-inlet approach is adopted to generate the prescribed PB while the subsequent wave evolution is determined by the RANS–VOF solver.

The analytical PB solution was imposed as a prescribed time-dependent boundary condition at the fixed inlet, rather than as a mechanical wavemaker signal. The inlet was located upstream of the focusing region to ensure that the wave packet entered the numerical flume in its pre-focusing stage, where the modulation remains relatively weak. In this study, the first-order NLSE solution was adopted, and second-order bound-wave corrections were not explicitly considered. Meanwhile, due to the limited computational time, the theoretically infinite PB waveform was truncated, while sufficient carrier-wave cycles were retained to reduce start-up effects and spectral leakage. These limitations have been considered in the interpretation of the results.

At the outlet of the numerical wave tank, a damping zone is employed for wave absorption. Within this zone, wave energy is progressively dissipated before the outgoing waves reach the downstream boundary, thereby reducing wave reflection, shortening the required computational-domain length and simulation time, and improving numerical stability and mesh adaptability. Consequently, the influence of reflected waves on the results is minimized. The modified momentum equation and damping source term are written as follows:

$${\displaystyle \frac{\partial \left(\rho w\right)}{\partial t}}+\nabla \cdot \left(\rho w\cdot \mathrm{u}\right)={\displaystyle \frac{\partial {\tau }_{xz}}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{yz}}{\partial y}}+{\displaystyle \frac{\partial {\tau }_{zz}}{\partial z}}+\rho f_z+S_z^d$$

(22)

$$S_z^d=\rho \left(f_1+f_2\left|w\right|\right){\displaystyle \frac{e^k-1}{e^1-1}}w$$

(23)

$$K=\left({\displaystyle \frac{x-x_{sd}}{x_{ed}-x_{sd}}}\right)$$

(24)

where x denotes the wave-propagation coordinate; x_sd and x_ed denote the starting and ending positions of the wave-damping zone; w is the vertical velocity component of the water particle; and f₁, f₂ are damping-model parameters.

This subsection summarizes the theoretical and numerical methods used to construct the two-dimensional numerical wave tank, including the governing equations, initial and boundary conditions, wave generation, and wave absorption.

3.2. Numerical Wave Tank Setup

This study is conducted using the two-dimensional numerical wave tank shown in Figure 1. The computational domain has a total length of L = L₁ + L₂ = 25 m, where L₁ denotes the main wave-propagation region and L₂ denotes the downstream wave-damping zone. The tank height is L₃ = 1.5 m, the out-of-plane thickness is y = 0.002 m to approximate a two-dimensional configuration, and the reference water depth is h = 1 m. The analytical PB free-surface elevation and velocity field are prescribed at the upstream boundary to generate the incident wave, which propagates in the positive x-direction. The top and downstream boundaries are treated as pressure outlets, while a damping zone is arranged immediately upstream of the downstream outlet to attenuate outgoing waves and reduce numerical reflection. The physical parameters adopted in the numerical wave tank are a water density of $\rho =1000$ kg/m³, a kinematic viscosity of $\upsilon =0.00001145$ m²/s, an atmospheric pressure of p₀ = 101000 pa, and a gravitational acceleration of g = 9.81 m/s².

3.3. Numerical Verification and Accuracy Analysis

To ensure accurate generation of Peregrine-breather-based rogue waves and to establish a reliable numerical basis for the subsequent investigation of their evolution over variable bathymetry, mesh and time-step sensitivity analyses were first conducted. For this study, the parameters of the underlying carrier wave are set as follows: wave amplitude a₀ = 0.02 m, wave period T = 0.8 s, wave length $\lambda =0.9984$ m, wave number k₀ = 6.29 m⁻¹, and water depth h = 1 m. These parameters satisfy the infinite water depth condition. The k₀h = 6.29 > 1.36 condition is required to ensure the sign of the nonlinear term in the NLSE, thereby guaranteeing the occurrence of MI [7,37]. The chosen parameter ${\epsilon }_0=a_0{k}_0=0.1258$ is slightly higher than the breaking threshold $\epsilon =0.12$ [38] of the first-order PB. The mesh size in the wave-height direction was varied from 0.003 to 0.0008 m, and the time step was varied from 0.002 to 0.0005 s, as detailed in

Table 1. Convergence validation of numerical wave tank.

Case	Grid Size in the Wave Height Direction (m)	Time Step (s)	Calculation Duration (s)
A1	0.003	0.0005	25
A2	0.0015	0.0005	25
A3	0.0008	0.0005	25
A4	0.0015	0.001	25
A5	0.0015	0.002	25

. The total simulation duration was 25 s.

Based on the analytical PB solution, the maximum crest is expected to occur at 12 m from the wave-generation boundary, corresponding to Gauge2. Monitoring points Gauge1 and Gauge3 are located at distances of 11 m and 13 m from the wave-generation boundary, respectively; monitoring point Gauge4 is located at a distance of 19 m from the wave-generation boundary (Figure 2). To evaluate the independence of the results from the computational mesh and time step, a convergence analysis was performed (Cases A1–A5). The results show that as the mesh is progressively refined and the time step is reduced, the numerical predictions converge and show excellent agreement with the theoretical solution (Figure 2). Although Case A3 employs a finer mesh, and Case A4 shows comparatively larger deviations in the background wave train, the predicted crest amplitudes in the principal focusing region remain close to the analytical solution. Considering both computational accuracy and cost, Case A2 was selected as the baseline configuration for the subsequent numerical wave-tank simulations because it requires substantially fewer grid cells than Case A3 while providing a better representation of the background wave train than Case A4.

4. Analysis of Peregrine Breather Propagation over Variable Bathymetry

This section focuses on the propagation characteristics of the PB over both uniform and non-uniform topographies. Based on the computational results from the numerical wave tank, a systematic analysis is conducted on the evolution of the wave packet’s surface profile, the variation in its focal position, and its energy during shoaling. Furthermore, the influence of topography on MI and the mechanisms of extreme wave generation are investigated. This study aims to provide theoretical support for constructing a more accurate forecasting system for nearshore extreme waves.

4.1. Temporal and Spatial Evolution Characteristics of Peregrine Breather Propagation

Figure 3 shows the computational wave tank, which has a total length of 25 m. A section with locally varying water depth is placed at different positions in the tank to represent typical bathymetric variations, such as nearshore slopes or submerged sand bars. In Figure 3, h = 1 m denotes the water depth, h_shelf denotes the water depth above the platform, L_p denotes the platform length in the composite bathymetric configuration, L₃ denotes the distance from the wavemaker boundary to the beginning of the bathymetric feature, L₄ denotes the size and position of the composite bathymetric feature, L₅ denotes the distance from the end of the bathymetric feature to the wave-damping zone, and L₆ denotes the length of the wave-damping zone. The wave inlet is located on the left, and a numerical wave-damping zone is placed on the right to reduce wave reflection. (Note: To further examine whether the local flow separation and vortex shedding induced by the sharp step edge significantly affect the free-surface wave evolution, an additional comparative case is included in Appendix B. In this case, the sharp step edge in the original bathymetry is replaced by a smooth gentle-slope transition, and the wave evolution characteristics under the two bathymetric configurations are compared.)

Wave gauges are arranged along the direction of wave propagation, as illustrated in Figure 3. Starting from the wave-generation inlet, the gauges are distributed at intervals of 0.5 m over 0~9 m, at a denser interval of 0.25 m over 9~20 m. This arrangement provides higher spatial resolution in the expected focusing region.

The numerical cases are summarized in

Table 2. Example cases of PB under different bathymetry and water-depth conditions.

Case	hshelf (m)	Lp (m)	L3 (m)	tf (s)	${\mathit{\eta }}_{\mathbf{max}} \mathbf{(}\mathit{m}\mathbf{)}$	Δtf (s)	xf (m)	Δxf (m)	k0hshelf
1	/	/	9	7.30	0.066	/	12	/	6.29
2	0.4	1	9	7.31	0.065	0.015	12	0	2.52
3	0.3	1	9	14.43	0.060	6.42	15.5	3.5	1.89
4	0.25	1	9	14.02	0.058	8.40	16.5	4.5	1.57
5	0.216	1	9	16.97	0.057	12.09	18.25	6.25	1.36
6	0.2	1	9	18.32	0.057	13.78	19	7	1.26
7	0.2	1	6	13.83	0.060	6.53	16.25	4.25	1.26
8	0.2	1	7	15.38	0.059	8.087	17.25	5.25	1.26
9	0.2	1	8	15.17	0.060	7.87	17	5	1.26
10	0.3	1	6	12.41	0.062	5.11	15.5	3.5	1.89
11	0.3	1	7	12.41	0.065	5.11	15.5	3.5	1.89
12	0.3	1	8	12.41	0.063	5.11	15.5	3.5	1.89
13	0.2	1.5	9	14.72	0.053	7.42	16.5	4.5	1.26
14	0.2	2	9	11.32	0.060	4.02	14.25	2.25	1.26
15	0.2	2.5	9	11.30	0.065	4	14.25	2.25	1.26
16	0.3	1.5	9	17.35	0.056	10.05	18.75	6.75	1.89
17	0.3	2	9	18.74	0.055	11.44	19.5	7.5	1.89
18	0.3	2.5	9	18.80	0.040	11.5	19.5	7.5	1.89
19	0.216	1.5	9	12.86	0.055	5.56	15.25	3.25	1.36
20	0.216	2	9	11.37	0.052	4.07	14.25	2.25	1.36
21	0.216	2.5	9	11.34	0.057	4.04	14.25	2.25	1.36

. To facilitate comparison, the PB parameters, including the background wave amplitude a = 0.02 m, wave period T = 0.8 s, and carrier wavenumber k₀ = 6.29 m⁻¹, are kept fixed, while only the bathymetric configuration is varied. The main variables are the water depth over the platform h_shelf, the platform length L_p, and the distance L₃ from the wavemaker boundary to the beginning of the bathymetric feature. Table 2 also lists the focusing time t_f, peak value of the focused wave ${\eta }_{\max }$, difference between the theoretical and numerical focusing times Δt_f, focusing position x_f, difference between the theoretical and numerical focusing positions Δx_f, and the size and position of the composite bathymetric feature.

Figure 4 presents the free-surface elevations recorded by wave gauges in the numerical wave tank and illustrates the evolution of the PB under different conditions, including the case without bottom topography, cases with bottom topography, and cases with different water depths. For the case without topography (Figure 4a), in the initial stage, the free-surface time histories at each measurement point exhibit nearly sinusoidal, narrow-banded quasi-monochromatic wave characteristics with relatively stable phase relationships. Subsequently, the initially weak MI then gradually amplifies during propagation in the numerical tank, consistent with the observations of Chabchoub et al. [16,17,18]. The wave envelope focuses progressively, and the amplitude growth at successive gauge locations reflects the continuous convergence of PB energy into a localized wave group governed by the group velocity. As the wave focusing intensifies, highly localized extreme wave crests and troughs emerge abruptly on the background wave train. The crest rises steeply over a short time interval and significantly exceeds the background wave amplitude. The maximum crest occurs at x_f = 12 m, t_f = 7.3s, ${\eta }_{\max }=0.0664$ m, corresponding to an amplification factor of approximately 3.3, which exceeds the theoretical factor of 3 [23]. Concurrently, adjacent waves develop peaked and asymmetric waveform profiles with strong transient energy concentration. After this peak, the wave train rapidly subsides and returns to the background wave amplitude, indicating that the PB exhibits pronounced spatiotemporal focusing that is well reproduced by the numerical wave tank simulations.

Figure 4b illustrates the evolution process under the conditions of h_shelf = 0.3 m, L_p = 1 m (Case 3). Compared with the flat-bottom scenario (Figure 4a), the differences in the breather under complex bathymetry mainly manifest in three aspects: focusing location, focusing process, and waveform structure. In the absence of bathymetry, the modulation growth of the wave packet occurs more continuously along the propagation path, energy accumulates gradually, and the maximum crest forms near the original focusing location at 12 m before rapidly attenuating to the background carrier wave. However, under the influence of variable bathymetry, local variations in water depth alter both the water depth conditions and the wave dispersion characteristics, suppressing energy convergence near the original focusing area. As a result, the maximum focusing position does not occur at 12 m but shifts overall downstream and is delayed until approximately 15.5 m before reaching its peak. In addition, the main focusing peak and the neighboring wave trains become more asymmetric, and the post-focusing recovery to the background wave train is slower. Figure 4c presents the evolution process under the conditions of h_shelf = 0.2 m, L_p = 1 m (Case 6). Due to the influence of topography, the local water depth is shallower, and the topographic effect on the focusing waves is more pronounced. Continuous defocusing occurs within the interval of 12.75 m~14.5 m, resulting in a substantial downstream delay in the attainment of maximum PB focusing. This behavior is qualitatively consistent with the laboratory observations discussed in Appendix A.

As shown in Figure 5, the maximum crest height varies along the propagation direction as the PB travels over variable bathymetry. Overall, as h_shelf decreases, PB focusing exhibits a coupled trend of reduced crest height and delayed focusing, with an approximately linear trend. When waves propagate over topography, energy dissipation leads to a slight attenuation in wave height. As shown in the velocity and vorticity fields in Figure A1 of Appendix A, the abrupt bathymetric transition near the platform end induces unavoidable wave energy scattering, and the velocity of water particles is suppressed. The red dashed–dotted line in Figure 5 represents the crest evolution for h_shelf = 0.4 m and k₀h_shelf = 2.52. Under this condition, the water-depth variation has little influence on the focusing location of the breather, which is consistent with the without-bottom result. However, the bathymetric feature still affects the post-focusing evolution. After passing over the topographic feature, the wave height decreases rapidly as the wave re-enters deeper water, then recovers to match the flat-bottom case results at a distance of 14 m from the wave inlet. However, at 16 m from the wave inlet, the decay rate of the wave crest is slower than in the flat-bottom case, and a slight increase in crest height is observed. The above results demonstrate that when k₀h_shelf is relatively large, although the bathymetric variation has little influence on the PB focusing location, it can induce oscillatory variations in the post-focusing crest evolution, resulting in deviations from the corresponding theoretical prediction.

When h_shelf is reduced to 0.3 m, k₀h_shelf = 1.89, the variable bathymetry suppresses PB focusing. In the elevated-platform region (11 m–12 m), the crest height of the breather no longer increases but slowly decays, continuing until about 14 m from the wave inlet. Afterward, the wave resumes its original focusing process, generating a maximum crest at 15.5 m with an amplitude approximately three times that of the carrier wave. As h_shelf further decreases, the suppression effect on the breather becomes more pronounced. After passing over the platform top and a subsequent distance, the crest height undergoes significant attenuation. At approximately 14 m from the inlet, after traveling about two wavelengths in deeper water, the breather begins to recover its focusing evolution and eventually develops a large crest close to three times the carrier-wave amplitude. Whitham theory for nonlinear waves predicts that, in shallower water, long-crested waves become modulationally stable; therefore, MI decreases as kh decreases and vanishes when kh < 1.363 because the coefficient of the cubic nonlinear term in the NLS vanishes at this threshold [39]. However, the present results indicate that the presence of topography suppresses the focusing of breathers even when k₀h_shelf = 1.89, leading to delayed focusing of the breather. After the wave passes over the bathymetric feature and re-enters deeper water, the self-focusing tendency can recover. When k₀h_shelf < 1.363, such as in the case of h_shelf = 0.2 m where k₀h_shelf = 1.26, MI recovers after the breather propagates over the bathymetry into deeper water, and its disappearance is only temporary.

Figure 6 shows the relationship between the PB focusing position and h_shelf. When h_shelf = 0.4 m, the focusing peak shows almost no downstream shift. However, when h_shelf decreases to the range of 0.3 m~0.2 m, the focusing position is significantly delayed downstream. Specifically, for h_shelf = 0.3~0.2 m, the peak positions occur at 15.5 → 19 m, corresponding to a downstream shift of approximately $3.5~7\lambda$. The focusing position exhibits approximately linear dependence on h_shelf. Because the bathymetric configuration adopted in this study is relatively idealized, the linear regression shown in Figure 6 should be regarded as a preliminary empirical relationship rather than a universal law.

Figure 7 presents the relationship between the platform length L_p and the downstream focusing shift Δx_f of the breather for different values of h_shelf. The results show that when h_shelf = 0.3 m, corresponding to k₀h_shelf > 1.363, Δx_f increases with the step length. In contrast, when h_shelf = 0.2 m, where k₀h_shelf < 1.363, the maximum delay distance occurs under the condition P_L = 1 m and is larger than that observed for h_shelf = 0.3 m. When P_L = 1.5 m, the delay distance decreases instead of increasing. For P_L = 2 m and 2.5 m, although waves with extremely high peaks are generated at a distance of 14 m from the wavemaker, they are not associated with the coherent focusing evolution of the PB. Detailed explanations are provided in Figure 8 and Figure 9.

Figure 8, Figure 9 and Figure 10 show the free-surface time histories at the focusing position, corresponding to the locations of the maximum crests recorded by the wave gauges, for three water depths (h_shelf = 0.2 m, h_shelf = 0.216 m and h_shelf = 0.3 m) and different platform lengths (L_p).

Under the condition of h_shelf = 0.2, 0.216 m, where k₀h_shelf < 1.363 is below the MI threshold, when $L_p\approx \lambda$, waves can re-focus at x_f = 19, 18.25 m after passing over the bathymetric feature, forming a typical breather structure. However, as the platform length increases from 1 to 2.5 times the wavelength, the topographic modulation becomes stronger and disrupts the breather structure. Consequently, the position of the maximum wave crest no longer exhibits the characteristic features of a PB. By contrast, for h_shelf = 0.3 m (Figure 10), where k₀h_shelf > 1.363 is above the MI threshold, the focusing position is delayed as L_p increases: the focusing location shifts from 15.5 m to 19.5 m, indicating a significant downstream postponement of the focus.

The above conclusions indicate that under shallower water conditions (h_shelf = 0.2 m and h_shelf = 0.216 m, k₀h_shelf < 1.363), although MI is suppressed or dissipated, an interaction distance of approximately two wavelengths is required to completely disrupt the breather structure. Moreover, Figure 8c,d and Figure 9c,d show that despite the breakdown of the breather structure, an extremely large wave emerges in the immediate lee of the topography (the deeper water region after the topography), which is consistent with the phenomena observed in our experiment (The detailed experimental results can be found in Appendix A). Specifically, rogue waves frequently occur downstream of submarine peak structures, after waves pass over a raised seabed feature [32,39]. The crest height of this wave also reaches approximately three times the amplitude of the carrier wave. This is consistent with the findings of Trulsen et al. [32], which indicate that as unidirectional waves propagate from deep water to shallow water, local maxima in kurtosis and skewness occur near the shallow-water side of the slope, indicating a significantly increased probability of rogue-wave occurrence. In contrast, when h_shelf = 0.3 m and k₀h_shelf > 1.363, increasing L_p primarily increases the delay in the focusing position but does not destroy the breather structure.

To investigate how the location of the bathymetric feature affects breather focusing, Figure 11 shows the modulation effects of PB focusing by bottom topography at different interaction stages under two relative water depths (h_shelf = 0.2 m and h_shelf = 0.3 m). In the figure, the vertical axis represents the peak envelope amplitude ${\eta }_{\max }$ within each wave packet period, and the horizontal axis denotes the corresponding focusing position x_f. The elevated bathymetric features are, respectively, placed within the ranges of 6~9 m, 7~10 m, 8~11 m, and 9~12 m. Under the shallow-water condition of h_shelf = 0.2 m (Figure 11a), the degree of delayed focusing is influenced by the position of the bathymetry: as the bathymetry shifts rearward overall, the delaying effect on focusing becomes more pronounced. The refocusing positions successively appear at x_f = 16.25, 17.25, 17, and 19 m, while the focusing profile maintains the sharp peak structure characteristic of a breather. This trend indicates that, in shallower water, a later interaction between the bathymetric feature and the breather more effectively disrupts the MI of the breather envelope and weakens the effective MI gain. The coherent focusing process that would otherwise be completed upstream is, therefore, interrupted, forcing the wave packet to re-establish coherence farther downstream, leading to secondary focusing with mild attenuation.

By comparison, for h_shelf = 0.3 m (Figure 11b), the four curves rise almost synchronously within the interval of 12~16 m and converge to form a consistent main peak (the peak value is about 0.065) between 15~16 m, with their refocusing positions consistently concentrated around 15.5 m. These results indicate that as the relative water depth increases, the coherent evolution of the breather envelope becomes more stable. Bathymetric disturbances mainly produce minor variations in amplitude and local phase, without substantially changing the main refocusing position. Consequently, the breather exhibits stronger robustness to variations in bottom bathymetry. However, when the breather interacts with the bathymetry at an earlier stage, a secondary focusing event with mild attenuation may occur, as observed at 18 m in Figure 11b.

Based on the two-dimensional bathymetric profile adopted in this study, the PB wave packet successively propagates over a deep-water horizontal section, an upslope shoaling section, and a shallow-water horizontal section. Different from the constant-water-depth condition, the variable-depth topography causes the local water depth h(x), wavenumber k(x), group velocity c_g(x), and the coefficients in the finite-depth NLS equation to vary along the propagation direction. For a given incident angular frequency $\omega$, the local wavenumber is determined by the finite-depth dispersion relation ${\omega }^2=gk\mathit{tanh}(kh)$. Therefore, as waves propagate from the deep-water section to the shallow-water section, kh gradually decreases, which leads to spatial variations in the dispersion coefficient $\lambda (h)$, nonlinear coefficient $\upsilon (h)$, and shoaling coefficient $\mu (h)$. According to the Djordjević–Redekopp variable-coefficient NLSE [15], the evolution of the wave-packet envelope can be expressed as

$$i\mu (h){\displaystyle \frac{dh}{dx}}B+i({\displaystyle \frac{\partial B}{\partial \mathrm{x}}}+{\displaystyle \frac{1}{c_g}}{\displaystyle \frac{\partial B}{\partial t}})+\lambda (h){\displaystyle \frac{{\partial }^{2}B}{\partial t^2}}=\upsilon {\left|B\right|}^{2}B+\varpi Bf(x)$$

(25)

Here, $\lambda (h)$ represents the local dispersive effect, $\upsilon (h)$ denotes the third-order nonlinear modulation effect, and $\mu (h)dh/dx$ describes the shoaling modulation induced by water-depth variation. For the bathymetry considered in this study, dh/dx in the deep-water horizontal section; thus, the shoaling term is inactive, and the PB wave packet mainly evolves through the dispersion–nonlinearity balance under a constant-depth condition. Hopwever, in the upslope shoaling section, dh/dk ≠ 0, so the shoaling term begins to participate in the envelope evolution. Meanwhile, $\lambda (h)$ and $\upsilon (h)$ also vary as kh decreases, thereby breaking the original dispersion–nonlinearity balance.

From the perspective of physical mechanisms, finite-depth effects in the shallow-water region modify the nonlinear modulation capacity of the wave packet. When the nonlinear coefficient $\upsilon (h)$ decreases or its rate of variation increases, the third-order nonlinear focusing effect is weakened. At the same time, variations in the dispersion coefficient $\lambda (h)$ alter the dispersive spreading rate of the wave packet. Since PB focusing is essentially governed by the coordinated interaction between nonlinear phase modulation and dispersion, the passage of the wave packet over the upslope bathymetry disturbs this phase-matching relationship, causing the focusing position originally formed under flat-bottom conditions to shift. The wave packet must continue to propagate for a certain distance after entering the deep-water section before a new dispersion–nonlinearity balance can be re-established under the updated local water-depth condition. Therefore, the downstream shift in the focusing point observed in the present numerical simulations can be interpreted as a delayed focusing mechanism induced by the spatial variation and cumulative effects of the modulation coefficients caused by variable-depth bathymetry.

4.2. Wavelet Energy Transform

Due to the localized, transient, and highly nonlinear modulation characteristics of PB, the wavelet transform can accurately capture the time–frequency energy evolution trajectories [40,41,42,43], thereby intuitively revealing underlying physical mechanisms such as MI. Compared with the Fourier transform, which is limited by the assumption of global stationarity, the wavelet transform exhibits significant theoretical and methodological advantages in analyzing such nonlinear transient phenomena due to its excellent time-frequency localization properties. Therefore, the Morlet wavelet is adopted as the mother wavelet, and the continuous wavelet transform is used to perform time-frequency analysis of extreme-wave evolution under uniform water depth and composite bathymetric conditions. Figure 12, Figure 13 and Figure 14 display the wavelet spectra for Case 1, Case 6, and Case 15, respectively, under three different operational conditions. From Figure 12 and Figure 13, two distinct gaps can be observed before and after the formation of the maximum wave crest. These gaps can be explained by an energy transfer mechanism: to form the maximum wave crest without increasing the total energy, part of the energy must be transferred to the moment when the maximum crest occurs. This energy transfer process cannot be observed within linear theory and may explain the sudden occurrence of freak waves—a phenomenon consistent with the observations of Hu et al. [27].

Figure 12 shows that, in Case 1 without bathymetry, the wavelet time–frequency energy spectrum exhibits the typical characteristics of an undisturbed PB. Before focusing, the spectral content gradually redistributes between the carrier-frequency band and the neighboring sidebands. At the focusing instant, the energy becomes highly localized in both the time and frequency domains, with pronounced energy concentration around the main carrier frequency, accompanied by local spectral broadening toward higher frequencies and symmetric sideband peaks. This energy-localization feature is particularly evident at x = 12 m, indicating strong nonlinear spectral interaction between the carrier band and adjacent sidebands. As the propagation distance gradually increases to x = 17.5 m, the energy of the breather structure after focusing is gradually reabsorbed by the neighboring carrier waves. The energy envelope generally maintains a single-peak profile, showing only slight narrowing of the spectral broadening and a gradual decrease in energy magnitude. Throughout the process, the central frequency remains concentrated without noticeable dispersion, and the energy ratio between the sidebands and the main frequency band remains essentially unchanged. This evolution reflects the dynamic balance between nonlinear self-modulation and linear dispersion within the breather, and is consistent with the focusing mechanism of the PB driven by MI.

When the breather interacts with platform bathymetry (where $L_p=\lambda$, h_shelf = 0.2 m, as shown in Figure 13), the wavelet time–frequency spectrum reveals a pronounced bathymetry-induced modulation of the spectral evolution. At x = 8.5 m, after the breather enters the shallow-water region, the energy envelope compresses, and the central frequency of the energy spectrum shifts toward higher frequencies, with pronounced enhancement on the high-frequency side. Between x = 10.25 m and 17.25 m after the t bathymetric interaction, the spectral localization weakens, and the PB focusing process is locally suppressed, corresponding to a stage of defocusing. As propagation continues into the downstream deep-water region, the bathymetric delay effect diminishes, and the breather refocuses, generating an extreme wave at x = 19 m. Overall, the process involves high-frequency energy migration, defocusing, and delayed refocusing. This sequence of spectral changes illustrates how variable bathymetry modulates the propagation and focusing behavior of the breather.

When the platform length L_p increases to $2.5\lambda$ (as shown in Figure 14), the energy evolution of the wave before x = 12 m remains generally consistent with that in the no-topography case. However, as the platform length increases, the bathymetric interaction length becomes larger, resulting in a more sustained influence of the bathymetric feature on the PB wave packet. Compared with the wavelet spectrum of the PB without topography, the wavelet time–frequency spectrum in this case exhibits distinctly different characteristics: the spectral band near the carrier frequency shows local splitting, and the focusing region no longer appears as a single, continuous PB-type energy-focusing structure. Instead, a more pronounced local increase in energy occurs toward the high-frequency range, indicating the significant influence of strong nonlinear shoaling on the wave-packet structure.

At x = 12.75 m, the dominant spectral peak begins to split, and the wavelet-power distribution gradually becomes bimodal. After the maximum wave crest passes over the topography (x = 13.75 m), its original structure is strongly disrupted. In the wavelet spectra at the downstream measurement points, the focused energy packet splits into a distinct bimodal structure, where the primary peak is defined as the “parent wave” and the secondary peak as the “child wave.” Meanwhile, persistent sideband residuals remain near the carrier frequency, indicating that the original wave packet has decomposed into multiple relatively independent energy envelopes. The continuous enhancement of energy in the downstream main frequency band further suggests that the bathymetry-induced energy redistribution dominates the subsequent evolution.

As wave propagation continues, the bimodal feature in the energy spectrum does not completely disappear. The energy of the parent wave gradually dissipates, whereas the child wave continues to propagate and undergo modulation, with its proportion of high-frequency energy remaining higher than in the other cases. In particular, after the maximum crest occurs, the spectral content shows enhanced redistribution toward higher frequencies followed by a reduction in spectral intensity, which differs from the typical sideband characteristics formed during the natural evolution of an intact PB under undisturbed conditions. These observations indicate that the stronger bathymetric forcing associated with the longer platform substantially disrupts the coherent PB structure and reduces its ability to recover during subsequent propagation. (In addition to the wavelet analysis described above, we have also supplemented our study with Fourier spectral analysis. The detailed analytical process and results are provided in Appendix C).

5. Conclusions

This study systematically investigates the evolution of PB over variable bathymetry and identifies, for the first time, a bathymetry-induced delayed-focusing phenomenon. Under the flat-bottom reference condition, the maximum PB crest occurs at approximately x_f = 12 m. When the platform length is fixed at L_p = 1 m and the bathymetric position is fixed at L₃ = 9 m, the focusing position remains at x_f = 12 m for k₀h_shelf = 2.52. As the relative water depth decreases to 1.89, 1.57, 1.36, and 1.26, the corresponding focusing positions progressively shift downstream to approximately 15.5, 16.5, 18.25, and 19 m, respectively. These results demonstrate that the bathymetry-induced focusing delay becomes progressively more pronounced as the local relative water depth decreases.

Theoretically, when kh > 1.363, breathers may induce nonlinear focusing due to MI, potentially leading to extreme waves [44]. However, this study reveals that even within the range of 1.363 < kh < 1.89, topographic influence can significantly suppress the focusing process of breather waves, thereby causing a delayed focusing phenomenon. Specifically, as the breather wave train propagates into shallower water, the growth of its envelope is notably inhibited: both the MI-driven energy transfer to sidebands and the phase-synchronization process are weakened. Above the region of topographic elevation, the wave train exhibits defocusing characteristics (energy becomes more spatially dispersed, and the growth of wave crests is constrained). After the wave train crosses the topography and returns to deeper water, nonlinear interactions intensify once again, accompanied by energy transfer toward higher frequencies. As the balance between dispersion and nonlinearity is gradually restored, the breather wave reasserts its typical focusing evolution characteristics and reaches its maximum crest at a further downstream location. This phenomenon, where the overall focusing position is delayed due to topographic effects, is referred to as “delayed focusing.” This finding is consistent with our earlier observations of breathers under random wave backgrounds over non-uniform bottom topography, as detailed in Appendix A. It should also be noted that the PB-like localized wave events observed in our previous experiments [34] emerged spontaneously within a JONSWAP irregular-wave field rather than being generated directly from a prescribed deterministic PB signal. Under such irregular-wave conditions, the downstream re-focusing behavior observed in the experiments may have been influenced by both variable bathymetry and stochastic interactions among the surrounding wave components, making it difficult to isolate the intrinsic effect of bathymetry on PB evolution. In contrast, the present CFD simulations prescribe a deterministic PB on a uniform monochromatic carrier-wave background, thereby excluding the random variability inherent in the irregular-wave field and allowing the influence of variable bathymetry to be examined more directly. The numerical results show that PB focusing is locally suppressed over the elevated bathymetric region and subsequently recovers farther downstream after the wave packet re-enters deeper water. This provides clearer numerical evidence that variable bathymetry itself can induce delayed focusing of a deterministic PB. Therefore, the previous irregular-wave experiments and the present deterministic-PB simulations provide complementary evidence, jointly indicating that the influence of variable bathymetry on the evolution of localized extreme waves can persist downstream of the bathymetric transition.

When k₀h_shelf > 1.363: As the platform length L_p increases, the focusing position of the breather shifts progressively downstream, resulting in a delayed focusing process. Although the breather’s fundamental envelope structure remains identifiable under the influence of the topography, this indicates that within this water depth range, the topographically induced local defocusing and phase disturbances do not completely suppress the development of MI. The presence of topography causes the breather, which would otherwise focus at a specific location, to be continuously perturbed and disrupted throughout the platform L_p, thereby shifting its focusing position downstream and ultimately leading to a delay effect.

When k₀h_shelf < 1.363: Once the platform length reaches approximately $1.5\lambda$, the breather structure begins to undergo progressive disruption, characterized by a loss of envelope symmetry, reduced organization of sidebands, and discontinuous evolution of the wave packet peak. When the L_p reaches about $2\lambda ~2.5\lambda$, the breather undergoes significant splitting and tends toward complete destruction. Notably, even after the breather is disrupted, relatively large wave peaks may still emerge behind the topography. However, these peaks are not a result of the typical MI focusing mechanism of the breather but are more likely attributable to strong nonlinear effects induced by shoaling.

The position of a composite uniform topography has a minor impact on the “delayed focusing” effect. Regardless of whether the topography is located in the growth stage just before the breather’s expected focusing or in the decay stage after focusing, as long as the geometric parameters of the topographic feature, namely the platform height and platform length, and the incident carrier wave and water depth conditions remain consistent, the overall shift in focusing position (the strength of the delay effect) remains largely unchanged. Only minor differences are observed in local wave train details, such as a slight attenuation in the wave peak value.

The present results further extend previous studies of Peregrine breathers (PBs) conducted under uniform deep-water and finite-depth conditions, broadening the existing understanding of PB evolution under uniform-depth conditions to the propagation, delayed focusing, and structural stability of deterministic PBs over spatially varying bathymetry. From a geophysical perspective, the present results indicate that the influence of variable bathymetry on a localized nonlinear wave packet is not necessarily confined to the region of water-depth variation itself. Spatial variations in water depth along the propagation direction can continuously modify the local wavenumber, group velocity, dispersion characteristics, nonlinear strength, and phase evolution, thereby perturbing the dispersion–nonlinearity balance governing PB self-focusing. This behavior is qualitatively consistent with previous studies of variable bathymetry. As shown by the results in Appendix A, and as discussed in greater detail in Ref. [34], the nonlinear dynamical response of a wave field induced by water-depth variation may persist beyond the local bathymetric transition, and a finite downstream adjustment distance may be required before the wave field adapts to the new water-depth environment. In natural environments such as continental-shelf transitions, shoals, and other non-uniform seabed regions, these results suggest that the region of strongest nonlinear wave amplification may not necessarily coincide with the shallowest location or the local bathymetric transition, but may occur farther downstream. These findings provide a useful reference for nearshore hydrodynamic research and for disaster prevention and mitigation related to extreme-wave hazards.

This study investigates the interaction mechanisms between variable bathymetry and breather-wave propagation through idealized seabed configurations, including both an abrupt step-shaped platform and a supplementary smooth-slope case. The results provide mechanistic insight into how bathymetric variation may influence the evolution, focusing, and local amplification of breather-type waves. However, because idealized geometries cannot fully represent the diversity of natural seabed topographies, the engineering implications should be interpreted as qualitative scientific references rather than as direct predictive guidance for all nearshore extreme-wave risk zones. Building upon our preliminary research [34,45] and integrating the works of Zhang J. and Benoit M., Tang et al., and Lyu et al. [46,47,48], we further clarify the regulatory role of variable topography in the generation of extreme waves. This lays the theoretical groundwork for investigating the propagation and evolution of PB over composite topography. Furthermore, the findings of this study are expected to inspire analogous research on weakly nonlinear wave phenomena in non-uniform media in other fields of physics.