Water Wave Model Research Articles

New Soliton Solutions and Trajectory Equations of Soliton Collision to a (2+1)-Dimensional Shallow Water Wave Model

New Soliton Solutions and Trajectory Equations of Soliton Collision to a (2+1)-Dimensional Shallow Water Wave Model

Interaction of lump, periodic, bright and kink soliton solutions of the (1+1)-dimensional Boussinesq equation using Hirota-bilinear approach

In this paper, we explore the characteristics of lump and interaction solutions for a (1+1) dimensional Boussinesq equation. By employing the Hirota bilinear method, we derive and analyze the exact solutions of this equation. Specifically, we achieve the lump with bright-bright soliton solution, 1-lump,2-lumps and 3-lumps with single bright soliton solution, lump with periodic, kink, and anti-kink soliton solutions. Alongside deriving these solutions, we also illustrate their dynamic properties through graphical simulations. The Boussinesq equation holds significant importance due to its applications in various domains, such as water wave modeling, coastal engineering, and the numerical simulation of water wave dynamics in harbors and shallow seas. Our research shows that the employed method is straightforward, easy to understand, and highly efficient, providing valuable insights into the equation’s nature and its practical applications.

Symmetric structures and dynamic analysis of a (2+1)-dimensional generalized Benny-Luke equation

We study the symmetric structures and dynamic analysis of a (2 + 1)-dimensional generalized Benny-Luke (GBL) equation based on the Lie symmetry method, the GBL equation is an important non-integrable model of water waves. Specifically, we construct multiple exact solutions of the GBL equation and obtain its nonlocally related systems. Firstly, the Lie point symmetries and conservation laws of the GBL equation are computed, and then we get the reduced ordinary differential equation from one of the conservation laws. Multiple methods, for example, the dynamical systems method, the power series method, the homogeneous balancing method and generalized variable separation method, are used to solve the ordinary differential equation and abundant exact solutions of the GBL equation are got. Finally, we extend these exact solutions by discrete symmetries, and give three-dimensional graphs of partial exact solutions. In addition, we construct the nonlocally related PDE systems, which contains the potential systems from the conservation laws and an inverse system from a Lie point symmetry of the GBL equation. These findings reveal the dynamical behavior behind the GBL equation and broaden the range of nonlinear water wave model solutions.

Exploring Soliton Solutions for Fractional Nonlinear Evolution Equations: A Focus on Regularized Long Wave and Shallow Water Wave Models with Beta Derivative

Exploring Soliton Solutions for Fractional Nonlinear Evolution Equations: A Focus on Regularized Long Wave and Shallow Water Wave Models with Beta Derivative

Abundant solitary wave solutions to the new extended (3+1)-dimensional nonlinear evolution equation arising in fluid dynamics

Nonlinear evolution equations appear in various branches of physics, including fluid dynamics, solid mechanics, quantum mechanics, and other fields. They are essential for describing systems where interactions and nonlinearity play a significant role, providing a more accurate representation of real-world phenomena than linear equations in certain cases. The study of nonlinear evolution equations involves exploring their solutions, stability properties, and the behavior of the systems they describe over time. In this study, we provide plenty of analytical solutions to a very recently proposed water wave model using the modified generalized exponential rational function, modified extended tanh-function, and the [Formula: see text]-expansion methods. To examine the dynamic characteristics of these results, we displayed three-dimensional (3D), contour, and two-dimensional (2D) visual representations of specific solutions. The outcomes reveal a multitude of novel solutions, emphasizing the robustness and effectiveness of the applied methodologies. It is important to note that all the solutions derived in this study are original and have not been previously documented in the existing literature. These novel solutions hold significant value for researchers in the realms of fluids and plasma physics, providing insights into the dynamics of nonlinear waves within various physical systems. Furthermore, this research contributes to a deeper understanding of the nature of nonlinear waves prevalent in seas and oceans, offering valuable knowledge for the broader study of such phenomena.

Particle trajectories in the KP-II equation

In the current work, we consider particle trajectories beneath traveling soliton solutions described by the Kadomtsev–Petiviashvili-II (KP-II) equation, which is a model for small-amplitude water waves in shallow water. The KP-II equation has a large number of exact solutions describing crossing line solitons. Here, we describe the particle drift induced by the single line soliton and various two- and three-soliton solutions on a two-dimensional horizontal domain.First, the derivation of the KP-II equation is used to exhibit expressions for the three components of the fluid velocity vector induced by a surface wave profile given by the KP-II equation. These velocity components can be used to specify a coupled system of ordinary differential equations (ODEs) which describe the motion of a fluid particle. Given an initial position inside the fluid for a specific particle, as well as continuously providing time-dependent values for the surface deflection, the ODE system can be solved numerically to find the particle path induced by the passage of a wave. A special numerical integration grid tracking the particle is used to handle integral terms occurring in the fluid velocity expressions.

Dynamics of resonant soliton, novel hybrid interaction, complex N-soliton and the abundant wave solutions to the (2+1)-dimensional Boussinesq equation

The presented work concerns with some novel solutions of the (2+1)-dimensional Boussinesq equation (BE), which acts as an important model for shallow water wave. Some resonant soliton solutions such as the X-shape soliton (XSS) and Y-shape soliton (YSS) solutions are developed via imposing the resonant conditions on the N-soliton solutions (N-SSs) that developed by the Hirota bilinear approach(HBA). Based on the XSS and YSS solutions, the novel hybrid interactions including the interaction between the 1-soliton and the YSS, the interaction between two YSS solutions are extracted. In addition, the complex N-SSs are also explored and discussed. Finally, the travelling wave solutions including the bright solitary and kinky solitary wave solutions are studied by employing the Bernoulli sub-equation function method (BSEFM). The graphs of the attained solutions are drawn to show the physical properties. The findings of this study are expected to help us apprehend the dynamics of the (2+1)-dimensional BE better.

A toy model for damped water waves

We consider a toy model for a damped water waves system in a domain [Formula: see text]. The toy model is based on the paradifferential formulation of the water waves system derived in the work of Alazard–Burq–Zuily [On the water-waves equations with surface tension, Duke Math. J. 158 (2011) 413–499]. The form of damping we consider is a modified sponge layer proposed for the [Formula: see text] water waves system by Clamond et al. in [An efficient model for three-dimensional surface wave simulations. Part II: Generation and absorption, J. Comput. Phys. 205 (2005) 686–705]. We show that, in the case of small Cauchy data, solutions to the toy model exhibit a quadratic lifespan. This is done via proving energy estimates with Klainerman vector fields.

Advancing computational models for wave dynamics applications with quartic trigonometric tension b-spline techniques

This paper presents a computational model using Quartic Trigonometric Tension B-spline (QT-TB) function with collocation approach for nonlinear dispersive wave equation. Shallow water wave modeling plays a crucial role in various fields, including coastal engineering, oceanography, and natural hazard assessment. The QT-TB function is employed for spatial derivatives, integrating the nonlinear term through the linearization technique. While maintaining most characteristics of traditional polynomial B-splines, this method improves numerical solutions, enhancing accuracy in modeling. The performance of the method is rigorously assessed across three benchmark problems, with results compared to those of prior studies employing identical parameters. Detailed numerical illustrations are presented. Graphical representations are utilized to illustrate the single solitary wave motion, dynamics of coupled solitary waves and dynamics of triplet solitary waves in this study. Numerical experiments validate the method's accuracy and efficiency in capturing RLW-driven wave dynamics, establishing it as a reliable tool for various applications. The presented work includes an analysis of the stability of the proposed scheme, employing the Fourier method and shows that its unconditional stable.

Analysis of perturbed Boussinesq equation via novel integrating schemes.

To analyze and study the behaviour of the shallow water waves, the perturbed Boussinesq equation has acquired fundamental importance. The principal objective of this paper is to manifest the exact traveling wave solution of the perturbed Boussinesq equation by two well known techniques named as, two variables [Formula: see text] expansion method and generalized projective Riccati equations method. A diverse array of soliton solutions, encompassing periodic, bright solitons, singular solitons and bright singular solitons are obtained by the applications of proposed techniques. The constraint conditions for newly constructed solutions are also specified. To enhance comprehension, the numerical illustrations of constructed solutions have been represented using surface plots, 2D plots and density plots. The results delineated in this paper transcend existing analysis, offering a novel, well-structured, and modern perspective. The solutions obtained not only enrich understanding of shallow water wave models but also exhibit efficacy in providing detailed descriptions of their dynamics.

The long wavelength limit of periodic solutions of water wave models

AbstractThe present essay is concerned with providing rigorous justification of a long‐standing practice in numerical simulation of partial differential equations. Theory often sets initial‐value problems on all of or . If the initial data are localized in space, it has been usual practice to approximate the problem by an associated periodic problem or a homogeneous Dirichlet problem set on a finite interval. While these strategies are commonplace, rigorous justification of the practice is sparse. It is our purpose here to indicate justification of this practice in the concrete context of a surface water wave model. While the theory worked out here is specific to the particular partial differential equation, it will be apparent to the reader that more general results may be derived using the same approach.

Analytical insights into the behavior of finite amplitude waves in plasma fluid dynamics

This study introduces innovative analytical solutions for the [Formula: see text]-dimensional nonlinear Jaulent–Miodek ([Formula: see text]) equation, a governing model elucidating the propagation characteristics of nonlinear shallow water waves with finite amplitude. Employing analytical methodologies such as the Khater II and unified methods, alongside the Adomian decomposition method as a semi-analytical approach, series solutions are derived with the primary aim of elucidating the fundamental physics dictating the evolution of [Formula: see text] waves. Within the realm of nonlinear fluid dynamics, the [Formula: see text] equation encapsulates the behavior of irrotational, inviscid, and incompressible fluid flow, wherein nonlinear effects and dispersion intricately balance to yield stable propagating waves. This equation encompasses terms representing nonlinear convection, dispersion, and nonlinearity effects. The analytical methodologies employed in this investigation yield solutions for various instances of the [Formula: see text] equation, demonstrating convergence, accuracy, and computational efficiency. The outcomes reveal that the Adomian decomposition method yields solutions congruent with those obtained through analytical techniques, thereby affirming the precision of the derived solutions. Furthermore, this study advances the comprehension of the physical implications inherent in the [Formula: see text] equation, serving as a benchmark for evaluating alternative methodologies. The analytical approaches elucidated in this research furnish accessible tools for addressing a diverse array of nonlinear wave equations in mathematical physics and engineering domains. In summary, the introduction of novel exact and approximate solutions significantly contributes to the advancement of knowledge pertaining to the [Formula: see text]-dimensional [Formula: see text] equation. The ramifications of this research extend to the modeling of shallow water waves, offering invaluable insights for researchers and practitioners engaged in the field.

Dynamical analysis and extraction of solitonic structures of a novel model in shallow water waves

In this study, we utilized a novel auto-Bäcklund transformation and the extended transformed rational function approach to analyze the extended reduced Jimbo–Miwa equation, a prominent equation within the KP hierarchy. The homogenous balance technique was employed to derive the auto-Bäcklund transformation of the equation, leading to the extraction of new exact solutions exhibiting solitary patterns. Additionally, we applied the extended transformed rational function method, which relies on the Hirota bilinear form of the governing equation, to generate complexiton solutions. Furthermore, we included 3D graphics visualizing the obtained solutions.

Sensitivity and wave propagation analysis of the time-fractional (3+1)-dimensional shallow water waves model

This study investigates the exact solutions of the time-fractional (3+1)-dimensional combined Korteweg–de Vries Benjamin–Bona–Mahony (KdV-BBM) equation. The considered model describes the long surface gravity waves of small amplitude, which portrays the two-way propagation of waves. The modified generalized Kudryashov method and the exp(-φ(ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi (\\xi $$\\end{document}))-expansion methods are employed to resolve the aforementioned issue because of their effectiveness and simplicity. For generality, the time fractional version is studied; more advanced solutions that do not exist in the literature were obtained. As a result, a variety of the exact wave solutions of the conformable (3+1)-dimensional KdV-BBM equation are obtained. The dynamical behaviors of some obtained solutions are represented with the proper parameter values. The used methods yield noteworthy results in obtaining the analytical solutions of fractional differential equations under various conditions. Besides, the sensitivity of regarding dynamical system is assessed to show the numerical stability effects.

Well-posedness of short time solutions and non-uniform dependence on the initial data for a shallow water wave model in critical Besov space

Well-posedness of short time solutions and non-uniform dependence on the initial data for a shallow water wave model in critical Besov space

Bright traveling breathers in media with long-range nonconvex dispersion.

The existence and properties of envelope solitary waves on a periodic traveling-wave background, called traveling breathers, are investigated numerically in representative nonlocal dispersive media. Using a fixed-point computational scheme, a space-time boundary-value problem for bright traveling breather solutions is solved for the weakly nonlinear Benjamin-Bona-Mahony equation, a nonlocal, regularized shallow water wave model, and the strongly nonlinear conduit equation, a nonlocal model of viscous core-annular flows. Curves of unit-mean traveling breather solutions within a three-dimensional parameter space are obtained. Resonance due to nonconvex, rational linear dispersion leads to a nonzero oscillatory background upon which traveling breathers propagate. These solutions exhibit a topological phase jump and so act as defects within the periodic background. For small amplitudes, traveling breathers are well approximated by bright soliton solutions of the nonlinear Schrödinger equationwith a negligibly small periodic background. These solutions are numerically continued into the large-amplitude regime as elevation defects on cnoidal or cnoidal-like periodic traveling-wave backgrounds. This study of bright traveling breathers provides insight into systems with nonconvex, nonlocal dispersion that occur in a variety of media such as internal oceanic waves subject to rotation and short, intense optical pulses.

Multiple solutions and conserved vectors of a shallow water wave equation arising in fluid mechanics; Lie group analysis

Investigation has shown that shallow water wave model equations constitute a class of hyperbolic partial differential equations describing the movement below a pressure surface in a fluid, and this makes them highly useful to be studied. Therefore, in this paper we analytically investigate a one-dimensional shallow water wave equation that is highly applicable in fluid dynamics. We leverage the robust nature of the Lie group technique to provide symmetries to the equation. Commutator and adjoint representations of symmetries were generated and presented in tabular form. Sequel to that, we perform reductions of the shallow water wave model under study via its point symmetries so that its exact solutions can be found. Thus, to achieve this, various combinations of the symmetries are made. Besides, we examine the travelling wave solutions of the model. The direct integration method is used to obtain elliptic integral functions as well as elliptic solutions in terms of the Weierstrass function. In addition, we utilize standard techniques such as (G′/G)-expansion, power series, and Kudryashov’s approaches to generate more closed-form solutions to the model. Other interesting solutions achieved include hyperbolic, trigonometric, rational, and exponential functions. In search of the physical meaning associated with the results, graphical demonstrations are included, and these are given in 2D, 3D, and density plots. Thereafter, the construction of conserved vectors for the aforementioned model is done with the use of the standard multiplier approach and Ibragimov’s theorem. These techniques give the opportunity to furnish various forms of conserved currents.

Mathematical modeling and component generalization of (2+1)-dimensional Schwarz–Kortweg–de Vries model in shallow water waves

This work employs the sub-ODE method to compute new soliton solutions for the component generalization of [Formula: see text]-dimensional Schwarz–Kortweg–de Vries (SKdV) equation in shallow water waves. The rational soliton solutions (RSS), Jacobian elliptic function (JEF), periodic solutions (PS), hyperbolic function solutions (HFS), positive solitons, dark soliton solutions (DSS) and bright soliton solutions (BSS) are constructed with this technique. Solitons-like solutions, HFS and trigonometric-function solutions (TFS) are also found as the JEF’s parameter m approaches values of 1 and 0, respectively. The suggested approach for the nonlinear equations in mathematical physics is concise, simple, effective, and promising.

Evolution of multi-solitons and interaction behaviors of lump to a (2+1) dimensional generalized shallow water wave model

In this paper, the trajectory equations of 1-lump before and after collision with high-order solitons and the degradation of some novel breather waves are studied in the (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation(gCBS). Firstly, we derive N-solitons for the gCBS equation by the Hirota bilinear form. With the help of N-solitons, we obtain M-lump as well as high-order breather based on the long-wave limit technique and the parametric conjugate method. Secondly, we construct many hybrid waves, such as the hybrid wave between breather and lump. Thirdly, the interaction phenomenon of lump-N-solitons(N → ∞) is investigated, and the theory of its existence is given and proved. Besides, the different degeneracies of double and single breather are discussed. Finally, we also present a large number of two-dimensional and three-dimensional images to better illustrate these nonlinear evolutionary behaviors.

A Study of Extreme Water Waves Using a Hierarchy of Models Based on Potential-Flow Theory

The formation of extreme waves arising from the interaction of three line-solitons with equal far-field amplitudes is examined through a hierarchy of water-wave models. The Kadomtsev–Petviashvili equation (KPE) is first used to prove analytically that its exact three-soliton solution has a ninefold maximum amplification that is achieved in the absence of spatial divergence. Reproducing this ninefold maximum paves the way for simulations based on both the Benney–Luke equations (BLE) and more advanced potential-flow equations (PFE). To preserve (for the sake of computations) the region of interaction, exact KPE solutions on an infinite domain are used to yield initial conditions that seed the BLE and PFE models within a periodic domain. The above strategies are realised by directly implementing the corresponding time-discretised variational principles within the finite-element environment Firedrake, one aim being automation of the generation of the algebraically cumbersome weak formulations. In the case of three-soliton interactions, it is found numerically that an amplification factor in the interval circa (7.6, 9) can be achieved within the BLE framework, whereas in the PFE framework, this falls to circa 7.8.