Advanced semi-analytical techniques for fractional shallow water wave models through the analogical structure of generalized ϕ-Caputo derivative operators.

This work develops two hybrid semi-analytical methods, namely the expansion new iterative method (ENIM) and the expansion homotopy perturbation method (EHPM), for solving the fractional Whitham-Broer-Kaup equations (WBKEs) involving the Erdélyi-Kober (EK) fractional derivative, since EK derivative is adopted due to its ability to incorporate scaling properties and nonlocal memory effects in a more flexible framework. The study focuses on the mathematical behavior of a fractional extension of the WBK system related to shallow water wave modeling. Several analytical properties of EK operators and their action on fractional power series expansions are established to support the proposed frameworks. By combining EK fractional integration with nonlinear operator decomposition and power series representations, ENIM and EHPM provide approximate solutions to nonlinear fractional partial differential equations. The methods are applied to the fractional WBK system, and numerical results demonstrate good agreement with the classical solution in the benchmark case ([Formula: see text]). For fractional cases ([Formula: see text]), the results are approximate and model-dependent. The comparison indicates that EHPM yields smaller absolute errors than ENIM within the tested parameter ranges. The influence of the fractional order α on the solution behavior is also illustrated, showing a transition between diffusive and classical wave patterns. These findings highlight the effectiveness of the proposed methods in terms of numerical accuracy within the considered framework.

Abstract Nonlinear partial differential equations (PDEs) are essential for describing wave propagation, interaction, and steepening in areas such as shallow-water dynamics, plasma physics, nonlinear optics, and elasticity. The Camassa–Holm (CH), Degasperis–Procesi (DP), and Fornberg–Whitham (FW) equations, together with their modified versions, form an important class of nonlinear dispersive models that exhibit solitons, peakons, and shock-like behaviors relevant to real-world applications. These models not only provide mathematical richness through integrability and diverse solution structures, but also capture physically significant features such as rogue waves and localized energy transport. In this work, we employ the improvised collocation method (ICM) to study both the classical and modified forms of the CH, DP equations, and the modified form of FW equations. The proposed scheme takes advantage of spline smoothness and local support, while the modifications improve stability and accuracy in handling nonlinear wave phenomena. Numerical experiments confirm that the method provides reliable and efficient approximations, even in the presence of sharp gradients. Stability analysis further supports its robustness for long-time simulations. The results highlight that the modified cubic B-spline collocation method is a powerful computational tool for investigating nonlinear dispersive PDEs, offering both accuracy and efficiency in capturing the dynamics of solitons and peakons.

This work investigates the dynamics of water wave elevation induced by a landslide traveling horizontally at a steady speed over a porous seabed. The analysis is conducted using three linearized models: the fully dispersive water wave model, the shallow water wave model, and the weakly dispersive water wave model. The one-dimensional horizontal closed-form analytical solutions for the propagation of water waves induced by a solid landslide translating at a steady velocity are attained for the fully dispersive water wave model, followed by the asymptotic solution in the far-field for the linear fully dispersive model. Furthermore, for each of the shallow water wave model and the weakly dispersive water wave model, the formal integral relation of a coupled system for free-surface elevation and velocity potential is derived. Each model provides a distinct approximation of wave dynamics, ranging from fully dispersive effects to shallow water assumptions. Additionally, the resonant solution is examined with the complete leading-wave far-field solution. For each of the three models, the horizontal and vertical velocity profiles are obtained. The impact of the seabed porosity on the propagation of water waves is incorporated through appropriate modifications to the primary equations for the shallow water wave and weakly dispersive water wave. The models provide insights into the influence of porosity on wave propagation characteristics, including amplitude and dispersion effects. The findings reveal that the seabed porosity significantly influences the wave elevation and propagation patterns. Porosity tends to dampen the wave amplitude and alter the dispersion characteristics. These effects are most pronounced in all three models. The obtained results are validated against the scenario of an impermeable seabed for two cases (including an experimental one), offering a comprehensive understanding of the role of the seabed permeability in landslide-induced wave generation.

In this paper, we try to obtain the decay rate for the free waves Tα(t)f≔eit|∇|+|∇|αf in two dimension for α > 0. We will characterize the impact of α on the decay estimate. In fact, we shall show the decay rate of Tα(t)f is t−1 if α ≤ 2, while the decay rate of Tα(t)f becomes t−5/6 if α > 2. It is well known it is associated with the linearized gravity-capillary water waves equation when α = 3 and gravity-hydroelastic waves equation when α = 5. Hence, our results are very helpful to establish the global well-posedness of a series of water wave models such as hydroelastic waves equation.

Under certain conditions on parameters, the corresponding traveling wave system of a highly nonlinear shallow water wave model is a planar dynamical system which has two singular straight lines. Corresponding dynamical system is detailed by utilizing dynamical system methods and singular traveling wave theory techniques. Additionally, this work gives bifurcations of phase portraits. Then, the dynamical behavior is derived. Under some specified parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, peakons, periodic peakons and compacton solutions can be found.

Abstract The article introduces an innovative approach to the shallow water wave models, the modified Degasperis–Procesi Equation (MD–PE) and modified Camassa–Holm Equation (MC–HE) with a time‐fractional derivative. A functional matrix of Integration of a unique class of graph polynomials, such as the Complete graph's characteristic polynomial, is constructed to approximate the solution of the models. The Caputo sense addresses the fractional time derivatives in the MD–PE and MC–HE models. The conventional collocation points switch the models to a set of nonlinear algebraic equations. Newton's approach discovers the unknown coefficients of the characteristic polynomials, resulting in the novel Characteristic polynomial collocation method () solution. Despite the conventional perturbation approach, the procedure isn't contingent on certain presumptions or variable constraints. Some numerical instances serve, and their outcomes are juxtaposed visually in 2D and 3D setups. Compared to typical hybrid interpreters that produce series solutions, this methodology generates exact outcomes utilizing less computational power and in a more straightforward manner. We compute absolute and other error norms to confirm the efficacy of our suggested method.

In the present study, numerical solutions of the important type of the fifth-order Korteweg–de Vries (KdV) equation for modeling shallow water waves, namely, Kawahara equation, are investigated. With the solutions of the higher order Kawahara equation used in the modeling of shallow water waves, the behavior change of the wave with parameter changes is revealed. Unlike the classical numerical methods, two effective numerical methods are combined and used together. One of the components of the scheme is differential quadrature method (DQM) that approaches the derivative of the mesh points in the solution domain. In other words, space discretization is completed by using DQM. The other component is the Crank–Nicolson scheme, which is the effective type of the finite difference method. By using the advantages of the DQM and Crank–Nicolson scheme with effective linearization technique together, high accurate solutions are obtained. Four different test problems and their various forms are solved numerically. Error norms, central processing unit times, rate of convergence, and three invariants are calculated and reported.

The higher-order nonlinear Schrödinger equation (NLS) (Dysthe’s equation in the context of water waves) models the time evolution of the slowly modulated amplitude of a wave packet in physical systems described by dispersive partial differential equations (PDEs). These systems, of which water waves are a canonical example, require the presence of a small-valued ordering parameter so that a multi-scale expansion can be performed. However, often the resulting system itself contains this parameter. Thus, these models are difficult to interpret from a formal asymptotics perspective. This article describes a procedure to derive a parameter-free, higher-order evolution equation for a generic infinite-dimensional dispersive PDE with weak linear damping and/or forcing. This is achieved by placing the PDE in an infinite-dimensional Hilbert space and Taylor expanding with Fréchet derivatives. An attractive feature of this procedure is that it can be used in many different physical settings, including water waves, nonlinear optics and any dispersive system with weak dissipation or forcing and does not assume any additional structure to the governing PDE, for example its Hamiltonian nature. To complement this, two specific examples with accompanying symbolic algebra code are demonstrated that can be used as a template for other physical systems.

A higher-order nonlinear Boussinesq system with a time-dependent boundary delay is considered. Sufficient conditions are presented to ensure the well-posedness of the problem by utilizing Kato’s variable norm technique and the Fixed Point Theorem. More significantly, the energy decay for the linearized problem is demonstrated using the energy method.

Abstract In this article, a Generalized Calogero-Bogoyavlenskii-Schiff (CBS) equation
is studied, serving as an extended shallow water wave model in higher dimen sions. Firstly, utilizing the Bell polynomial method, the bilinear form of the
equation, bilinear B¨acklund transformation, Lax pair and infinite conservation
laws are derived, confirming the equation’s complete integrability in the context
of the Lax pair. Subsequently, the nonlinear superposition formula of the equa tion is constructed based on the derived bilinear B¨acklund transformation, and
an array of infinite superposition soliton solutions of the equation are formulat ed using this nonlinear superposition formula. Lastly, leveraging the obtained
bilinear equation, infinite superposition solutions of various functional types are
constructed, and their dynamic characteristics are analyzed through illustrated
solution images. It is noteworthy that this paper not only uncovers a multitude
of properties through the Bell polynomial method but also derives both infinite
linear and nonlinear superposition solutions, enriching the diversity of solutions,
these aspects have not been previously explored in existing literature.

The formation of rogue waves in coastal areas can be triggered by modulational instability (MI, known also as Benjamin-Feir instability), especially in deep water. The analysis of rogue wave generation and statistical characteristics of irregular waves in finite depth has predominantly focused on low-order (third-order) nonlinear interactions. In the present study, we conduct numerical simulations using a fully nonlinear, spectrally accurate water wave model (Klahn et al. 2021) to explore the statistical properties of irregular, uni-directional wave fields initially described by a TMA spectrum (Holthuijsen 2010). The numerical model is initially validated through the simulation of modulational instability in nonlinear plane wave trains in deep water. Subsequently, a series of random uni-directional wave fields, spanning a broad spectrum of water depths, is examined. This investigation encompasses various aspects of wave statistics, and discusses the relationship between the occurrence probability of extreme waves (rogue waves) with full nonlinearity. The significance of fully nonlinear behavior, as opposed to third-order nonlinearity, is also assessed.

Integrability characteristics and exact solutions of an extended $$(3+1)$$-dimensional variable-coefficient shallow water wave model

Waves on graphs are a current subject of research interest. As opposed to flows on graphs, the reflection–transmission of waves at the graph’s vertex is a problem that needs to be further modelled mathematically. The literature on the reflection and transmission of waves at a vertex is scarce. Some articles are reviewed and discussed. Water waves are a good topic for comparing different mathematical models, from hyperbolic conservation laws to weakly nonlinear, weakly dispersive systems of partial differential equations on a two-dimensional fattened (thick) graph and the respective one-dimensional graph-model reduction. In this study, we present a particular water wave model in which junction angles are systematically included in the mathematical model. Comparing the solutions with the fattened-graph model gave rise to a more general compatibility condition at the vertex. Current research topics of interest are outlined at the end.

Abstract This article investigates the traveling wave solution of the fractional (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili model by using the complete discriminant system method. These solutions not only include rational function solutions, trigonometric function solutions, but also Jacobian function solutions. In order to illustrate the propagation of these solutions in the field of nonlinear optics and water wave models, some three-dimensional, two-dimensional, and contour maps are drawn. Meanwhile, the phase portrait of two-dimensional dynamical systems and its perturbation systems are studied using the planar dynamical system analysis method. By drawing phase diagrams, it is easy to observe the stability, periodicity, and chaotic behavior of two-dimensional dynamical systems through geometric visualization, which can also provide strong basis for researchers to design corresponding control systems.

The Boussinesq equation is essential for studying the behavior of shallow water waves, surface waves in oceans and rivers, and the propagation of long waves in nonlinear systems. Its fractional form allows for a more accurate representation of wave dynamics by incorporating the effects of nonlocal interactions and memory. In this paper, we focus on obtaining exact traveling wave solutions for the space-time fractional Boussinesq equation using two well-established methods: the modified Sardar sub-equation method and the new extended direct algebraic method, both implemented with Atangana’s beta derivative. By applying these methods, we derive a variety of soliton solutions, including kink, anti-kink, periodic, dark, bright, and singular solitary waves. These solutions are presented in different mathematical forms, such as rational, hyperbolic, trigonometric, and exponential functions. This study not only provides new solutions but also enhances the understanding of wave propagation in fractional models, demonstrating the efficiency and applicability of the chosen methods. A comparative analysis of the methods and results is presented, along with an examination of the impact of fractional derivatives by adjusting their values. The study also includes 2D and 3D plots that illustrate the temporal behavior of the solutions. This study demonstrates that the methods employed are applicable to other nonlinear models in mathematical physics. A detailed analysis of the model’s behavior is conducted, focusing on bifurcation, chaos, and stability. Phase portrait analysis at critical points reveals shifts in the system’s dynamics, and introducing an external periodic force generates chaotic patterns. The solutions provided offer new insights into shallow water wave models, presenting effective tools for in-depth investigation of wave dynamics. All solutions are verified through MATHEMATICA and MATLAB simulations, ensuring their accuracy and reliability.

We study the recently-proposed hyperbolic approximation of the Korteweg-de Vries equation (KdV). We show that this approximation, which we call KdVH, possesses a rich variety of solutions, including solitary wave solutions that approximate KdV solitons, as well as other solitary and periodic solutions that are related to higher-order water wave models, and may include singularities. We analyze a class of implicit–explicit Runge–Kutta time discretizations for KdVH that are asymptotic preserving, energy conserving, and can be applied to other hyperbolized systems. We also develop structure-preserving spatial discretizations based on summation-by-parts operators in space including finite difference, discontinuous Galerkin, and Fourier methods. We use the entropy relaxation approach to make the fully discrete schemes energy-preserving. Numerical experiments demonstrate the effectiveness of these discretizations.

Beach groynes are structures for erosion protection along sandy coasts near inlets and can reduce the coastal erosion substantially, but open groynes cannot stop erosion completely because sand can be removed from the groyne compartments by cross-shore processes. Beach groynes should be designed with sufficient bypassing of sand to minimise erosion. Regular beach maintenance is required to keep a sufficient beach width for recreational purposes. The effectiveness of groyne compartments can be significantly improved by using T-head groynes or by using a submerged sill or breakwater in between the groynes. An economic evaluation shows that the beach maintenance costs over 50 years may be substantially higher than the construction costs of a submerged breakwater. An important parameter to be studied is the longshore transport, which requires detailed information of the wave climate, preferably based on measured data (offshore buoys) in combination with deep water wave modelling. Various models have been used to determine the net longshore sand transport and coastline changes. The design of groynes to reduce coastal erosion is illustrated by three field cases (Atlantic coast near Soulac, France; Lagos coast, Nigeria and Black Sea coast, Romania). These example cases show that beach groynes are effective structures, but sufficient bypassing of longshore sand transport is essential to minimise erosion. Regular beach fills in the groyne compartments may be required at high-energy (exposed) coasts. A submerged or emerged breakwater can be built between the groynes to protect the beach in the groyne compartments against erosion by cross-shore processes.

In this paper, we develop an efficient numerical method for obtaining numerical solutions of one-dimensional, two-dimensional, and three-dimensional regularized long wave equation which is a nonlinear partial differential equation and has applications in modeling of water waves. We use a time splitting algorithm based on Strang splitting for discretizing time variable of the considered problem. We also investigated linear stability analysis of time discrete scheme via von Neumann approach. Then for space discretization, barycentric rational interpolants of Floater–Hormann are employed with a linearization technique. By combining these time and space discretizations, finding numerical solution of considered partial differential equation is reduced to solving linear system of equations. Detailed numerical simulations are performed to assess accuracy of the developed method. Comparisons with many methods available in literature for considered partial differential equation demonstrate that the proposed method is accurate, efficient and feasible.

In this paper, we generate different types of solitary and peaked soliton traveling wave solutions for the Qiao equations, which are closely related to the propagation modeling of shallow water waves. An expansion method has been employed to achieve this. For certain special values of the nonlinearity term in the Qiao equation, solitary waves and peaked solitons are obtained. Subsequently, traveling wave solutions in hyperbolic form, distinct from the existing literature and dependent on the nonlinearity term, are produced for the most general case. Specifically, the responses of parameters representing the degree of nonlinearity and wave velocity for these traveling wave solutions are discussed. The main motivation of this work is to provide a better understanding of the phenomenon of nonlinear wave propagation and in this context to reveal the mathematical richness offered by nonlinear wave equations such as the Qiao equations. The aim is to elucidate the nonlinear wave propagation phenomenon through the generated traveling wave solutions, in accordance with the basic principles of shallow water theory. Additionally, the interactions between solutions are examined separately when the speed of the solitary wave exceeds that of the peaked soliton. It is hoped that this work will shed light on developments in the field of nonlinear dynamics.