Shallow Water Wave Equation Research Articles

Multiple localized waves to the (2+1)-dimensional shallow water waveequation on non-flat constant backgrounds and their applications

In this paper, a new general bilinear Bäcklund transformation and Lax pair for the (2+1)-dimensional shallow water wave equation are given in terms of the binary Bell polynomials. Based on this transformation along with introducing an arbitrary function, the multi-kink soliton, line breather, and multi-line rogue wave solutions on a non-flat constant background plane are derived. Further, we found that the dynamic pattern of line breather on the background of periodic line waves are similar to the two-periodic wave solutions obtained through a multi-dimensional Riemann theta function. Also, the generation mechanism and smooth conditions of the line rogue waves on the periodic line wave background are presented with long-wave limit method. Additionally, a family of new rational solutions, consisting of line rogue waves and line solitons, are derived, which have never been reported before. Furthermore, the present work can be directly applied to other nonlinear equations.

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.

Local-in-space wave breaking criteria for a shallow water wave equation

A shallow water wave equation including the famous Degasperis-Procesi model is considered. Firstly, the L 2 conservation law for the equation is derived. Secondly, using the methods of transport equation, we establish the boundedness of solutions for the shallow water wave equation. Finally, utilizing the approaches to construct Lyapunov functions, we find local-in-space wave breaking criteria of its solutions on the line R and circle S under certain restrictions on the coefficients of the equation.

Travelling-wave, Mixed-lump-kink and Mixed-rogue-wave-kink Solutions for an Extended (3+1)-dimensional Shallow Water Wave Equation in Oceanography and Atmospheric Science

Travelling-wave, Mixed-lump-kink and Mixed-rogue-wave-kink Solutions for an Extended (3+1)-dimensional Shallow Water Wave Equation in Oceanography and Atmospheric Science

Some Novel Fusion and Fission Phenomena for an Extended (2+1)-Dimensional Shallow Water Wave Equation

An extended (2+1)-dimensional shallow water wave (SWW) model which can describe the evolution of nonlinear shallow water wave propagation in two spatial and temporal coordinates, is systematically studied. The multi-linear variable separation approach is addressed to the extended (2+1)-dimensional SWW equation. The variable separation solution consisting of two arbitrary functions is obtained, by assumption, from a specific ansatz. By selecting these two arbitrary functions as the exponential and trigonometric forms, resonant dromion, lump, and solitoff solutions are derived. Meanwhile, some novel fission and fusion phenomena including the semifoldons, peakons, lump, dromions, and periodic waves are studied with graphical and analytical methods. The results can be used to enhance the variety of the dynamics of the nonlinear wave fields related by engineering and mathematical physics.

Resonant Y-type soliton, X-type soliton and some novel hybrid interaction solutions to the (3+1)-dimensional nonlinear evolution equation for shallow-water waves

This research aims to explore some novel solutions to the (3+1)-dimensional nonlinear evolution equation (NEE) for the shallow-water waves. The resonant Y-type soliton (YTS) and X-type soliton (XTS) solutions are derived by applying the novel resonant conditions on the N-soliton solutions(N-SSs) which are extracted via the Hirota bilinear approach. Additionally, some novel and interesting hybrid interaction solutions like the interaction between Y-type soliton and 1-soliton, interaction between Y-type soliton and 1-breather solution, interaction between the Y-type soliton and the soliton molecule on the (x, y)-plane, and interaction between the X-type soliton and 1-soliton are also ascertained. The dynamic attributes of the obtained solutions are described graphically to unveil their physical behaviors. The findings in this work can help us better apprehend the nonlinear dynamics of the considered equation.

A new high-order Camassa-Holm-type equation for shallow water waves moving over a shear flow

A new high-order Camassa-Holm-type equation for shallow water waves moving over a shear flow

Lump waves, bright-dark solitons and some novel interaction solutions in (3+1)-dimensional shallow water wave equation

Abstract The (3+1)-dimensional Geng equation is an extended version of the KdV model that describes the wave dynamics behavior of shallow water waves in complex applications. In this study, we discuss the (3+1)-dimensional Geng equation using the bilinear neural network method. By incorporating specific activation functions into the neural network model, new test functions are constructed. Using symbolic computational techniques and selecting appropriate parameters, we systematically obtain new meaningful exact solutions of some (3+1)-dimensional Geng equations, including dark lump solutions, three kinds of interaction solutions, and bright and dark soliton solutions. Furthermore, the results are visualized through diagrams of different categories, which intuitively demonstrate the evolution process and physical characteristics of the waves.

Studies on an extended (3+1)-dimensional variable-coefficient shallow water wave equation: Bilinear form, Pfaffian solutions and Solitonic interactions

Studies on an extended (3+1)-dimensional variable-coefficient shallow water wave equation: Bilinear form, Pfaffian solutions and Solitonic interactions

Local well-posedness and blow-up criterion to a nonlinear shallow water wave equation

<abstract><p>The initial data problem to a nonlinear shallow water wave equation in nonhomogeneous Besov space is discussed. Using the decomposition of Littlewood-Paley and the properties of nonhomogeneous Besov space, we establish the well-posedness of short time solutions for the equation in the Besov space. A blow-up criterion of solutions is also obtained.</p></abstract>

Soliton interactions, soliton bifurcations and molecules, breather molecules, breather-to-soliton transitions, and conservation laws for a nonlinear (3+1)-dimensional shallow water wave equation

Soliton interactions, soliton bifurcations and molecules, breather molecules, breather-to-soliton transitions, and conservation laws for a nonlinear (3+1)-dimensional shallow water wave equation

The dynamical property of a nonlinear shallow water wave equation with inhomogeneous boundary conditions

A nonlinear shallow water wave equation containing the Fornberg–Whitham model is considered. The phase portrait analytical technique is employed to establish the existence of the smooth, peaked and cusped solitary wave solutions of the equation under inhomogeneous boundary conditions. Asymptotic and numerical analysis illustrates the dynamical features for the smooth, peaked and cusped solitary wave solutions. Our results are helpful to further understand the dynamical tendency of the solutions when the space variable tends to positive or negative infinite.

Bragg reflection of linear shallow-water waves by an array of widely spaced trapezoidal bars on a fringing reef flat

A closed-form solution to the linear shallow-water wave equation for wave reflection by a finite periodic array of widely spaced trapezoidal bars on a fringing reef flat is presented. It is shown that, when the total number, height, or top width of the trapezoidal bars increases, the strength of the Bragg resonance increases but the downward shift of the resonance phase becomes more pronounced. When the width of the reef face increases or when the reef face changes from a concave profile to a convex profile, the strength of the Bragg resonance decreases and the downward shift of the resonance phase also becomes more pronounced. Especially, it is found that the location of trapezoidal bars plays an important role in the strength and occurrence condition of the Bragg resonant reflection. Finally, the optimal collocation of the location and width of trapezoidal bars is investigated. Considering both construction cost and reflecting capacity, the relative width is recommended to be within 0.4≤wb/d≤0.6 and the distance from the reef edge to trapezoidal bars is suggested to be equal to the spacing of trapezoidal bars.

Variational approach for (3+1)-dimensional shallow water wave equation

This paper focuses on the variational approach for solving the (3+1)-dimensional shallow water wave equation. Some families of soliton-type solutions and periodic wave solutions are given, and some remarks about the variational formulation for the special (2+1)-dimensional shallow water wave equation in [Gu, et al. Results in Physics, 45 (2023) 106213] are provided. Numerical results are presented to show the propagation behaviour.

Soliton molecules, interaction and other wave solutions of the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

In the present work, we aim to explore the new (3+1)-dimensional integrable fourth-order nonlinear equation(IFNE) for describing the shallow water waves. First, we study its N-soliton solutions via the bilinear form which is constructed by applying the Cole-Hopf transform. The resonance conditions of the soliton molecular are extracted and the soliton molecules are obtained. Second, the ansatz function method together with the symbolic computation, is implemented to develop the interaction wave solutions(IWSs). Finally, we take advantage of the Bernoulli sub-equation function method(BSFM) to look into the travelling wave solutions(TWSs). Different kinds of the TWSs like the singular-kink and kink solitary wave solutions are found. Correspondingly, the dynamic performances of the solutions are depicted graphically to present the physical interpretations. And for all we know, the solutions got in this work are all new and can be regarded as an extension of the solutions for the new (3+1) dimensional IFNE, which are expected to have practical significance for the application of these equations in physics.

A novel analysis of Cole–Hopf transformations in different dimensions, solitons, and rogue waves for a (2 + 1)-dimensional shallow water wave equation of ion-acoustic waves in plasmas

This work investigates a (2 + 1)-dimensional shallow water wave equation of ion-acoustic waves in plasma physics. It comprehensively analyzes Cole–Hopf transformations concerning dimensions x, y, and t and obtains the dispersion for a phase variable of this equation. We show that the soliton solutions are independent of the different logarithmic transformations for the investigated equation. We also explore the linear equations in the auxiliary function f present in Cole–Hopf transformations. We study this equation's first- and second-order rogue waves using a generalized N-rogue wave expression from the N-soliton Hirota technique. We generate the rogue waves by applying a symbolic technique with β and γ as center parameters. We create rogue wave solutions for first- and second-order using direct computation for appropriate choices of several constants in the equation and center parameters. We obtain a trilinear equation by transforming variables ξ and y via logarithmic transformation for u in the function F. We harness the computational power of the symbolic tool Mathematica to demonstrate the graphics of the soliton and center-controlled rogue wave solutions with suitable choices of parameters. The outcomes of this study transcend the confines of plasma physics, shedding light on the interaction dynamics of ion-acoustic solitons in three-dimensional space. The equation's implications resonate across diverse scientific domains, encompassing classical shallow water theory, fluid dynamics, optical fibers, nonlinear dynamics, and many other nonlinear fields.

Stability analysis and dynamics of solitary wave solutions of the (3+1)-dimensional generalized shallow water wave equation using the Ricatti equation mapping method

Our aim is to examine the dynamic characteristics of the (3+1)-dimensional generalized equation governing shallow water waves. When the horizontal extent of the fluid significantly surpasses the vertical dimension, the employment of shallow water equations becomes appropriate. By employing an inventive Ricatti equation mapping approach, we have obtained a range of solitary wave solutions in both explicit and generalized forms. Solitons are particularly useful in signal and energy transmission due to their ability to preserve their shape during propagation. By studying soliton solutions in nonlinear systems, we can gain valuable knowledge applicable to practical fields. To deepen our grasp of the equation’s physical implications, we have presented some solutions through 3D, 2D, and contour graphics. Employing a linear stability approach, we confirm the equation’s stability. The method used distinguishes out for its simplicity, dependability, and ability to generate novel solutions for nonlinear partial differential equations in the area of mathematical physics. The research findings reported here demonstrate the viability of the used method in studying nonlinear phenomena in the studied equation as well as other nonlinear problems in mathematical physics. By employing this tool, academics have the chance to deepen and increase their grasp of the complex mathematical concepts underlying real-world problems.

Exact Solutions of Nonlinear Partial Differential Equations Using the Extended Kudryashov Method and Some Properties

In this paper, we consider how to find new exact solutions for nonlinear partial differential equations using the extended Kudryashov method. This method mainly uses the Riccati equation and the Bernoulli equation where there are some underdetermined constant parameters. And we also use the concept of symmetry to study its reduction equation, Lie transformation group, self-adjointness, and conservation laws. This paper mainly studies the Boussinesq class and the shallow water wave equation in (1 + 1) dimensions and tries to find new exact solutions and symmetry properties of them.

A high-order domain preserving DG method for the two-layer shallow water equations

The bilayer shallow water wave equations in one-dimensional space are considered in this paper. The equations admit two groups of characteristic velocities, which are the first-order approximation of the eigenvalues. Due to the numerical instability, the characteristic velocities may become complex, and thus the system is not hyperbolic and yields to the so-called Kelvin–Helmholtz instability at the interface of the two layers. To overcome this issue, an invariant domain preserving DG method is presented for the bilayer shallow water wave equations. The proposed method is high-order accurate, conservative and can keep the characteristic velocities being real provided that the initial characteristic velocities are real. Therefore, the Kelvin–Helmholtz instability at the interface can be avoided. Representative numerical examples are chose to demonstrate the performance of the proposed method.

Soliton molecules, novel hybrid interaction solutions and periodic wave solutions to the (3 + 1)-dimensional nonlinear evolution equation for shallow-water waves

The central target of this research is looking into some novel solutions of the (3 + 1)-dimensional nonlinear evolution equation (NEE) for the shallow water waves. By manipulating the Hirota bilinear approach (HBA), the N-soliton solutions(N-SSs) are extracted. Then applying the resonance condition on the N-SSs, we successfully derive the soliton molecules (SMS) on the (x,y), (x,z) and (y,z)-planes. In addition, abundant novel hybrid interaction solutions are also studied by assigning the reasonable parameters, including the interaction between 1-soltion and the breather solution, interaction between 1-soltion and the soliton molecule(SM) with two solitons, interaction between two-breather solutions, interaction between breather solution and the SM of two solitons, interaction between two SMS of two solitons, interaction bwtween 1-soliton and the SM of three solitons. In the end, the periodic wave solutions (PWSs) are explored by taking advantage of the test function approach and symbolic calculation. The dynamic attributes of the obtained solutions are described graphically to unveil their physical behaviors. The discoveries in this work can enable us better understand the nonlinear dynamics of the considered equation.