Long Water Waves Research Articles

Symmetry Reductions of the (1 + 1)-Dimensional Broer–Kaup System Using the Generalized Double Reduction Method

The generalized theory of the double reduction of systems of partial differential equations (PDEs) based on the association of conservation laws with Lie–Bäcklund symmetries is one of the most effective algorithms for performing symmetry reductions of PDEs. In this article, we apply the theory to a (1 + 1)-dimensional Broer–Kaup (BK) system, which is a pair of nonlinear PDEs that arise in the modeling of the propagation of long waves in shallow water. We find symmetries and construct six local conservation laws of the BK system arising from low-order multipliers. We establish associations between the Lie point symmetries and conservation laws and exploit the association to perform double reductions of the system, reducing it to first-order ordinary differential equations or algebraic equations. Our paper contributes to the broader understanding and application of the generalized double reduction method in the analysis of nonlinear PDEs.

New insight to the shallow-water studies on a (2+1)-dimensional generalized Broer-Kaup system

Of current interest, there have recently appeared a series of the shallow-water papers in Mod. Phys. Lett. B. Illuminated by those papers and to make the story more complete, for the nonlinear long waves in the shallow water, this paper is scheduled to investigate a (2+1)-dimensional generalized Broer-Kaup system. As for the wave height and wave horizontal velocity, we employ symbolic computation, with the view of constructing out two groups of the similarity reductions.

Nonlinear long waves in shallow water for normalized Boussinesq equations

In this work, we investigate a partial differential equation (PDE) derived from a Boussinesq system of two equations that describe the wave motion in shallow water when the wavelength is small compared to wave amplitude. The Lie point symmetries of the PDE are utilized to reduce it to an ordinary differential equation (ODE) the travelling wave solutions of which are obtained. These solutions are of trigonometric, hyperbolic and exponential types. Graphical representation of some solutions exhibiting solitary and cnoidal wave phenomena are presented.

A SHORT SOLUTION OF THE LOCAL FRACTIONAL (2+1)-DIMENSIONAL DISPERSIVE LONG WATER WAVE SYSTEM

In this paper, a local fractional Riccati differential equation method is applied. A new travelling wave solution to the nonlinear local fractional [Formula: see text]-dimensional dispersive long water wave system is investigated. After travelling wave transformation, the governing model studied is converted into nonlinear ordinary differential equation. Some properties with the strain conditions are also reported.

The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method

<p>The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology.</p>

The Efficient Method to Solve the Conformable Time Fractional Benney Equation

The current study employed the innovative conformable fractional method to analyze the nonlinear Benney equations involving the conformable fractional derivative. Conformable fractional Benney equations have been examined by the conformable q‐Shehu analysis transform method. By including nonlinear factors, it offers a more precise depiction of wave propagation compared to linear models. Various natural phenomena, including ocean waves, plasma waves, and some forms of solitons, display nonlinear behavior that cannot be precisely explained by linear equations. The fractional Benney equation is important because it extends the classical Benney equation, which describes the evolution of weakly nonlinear and weakly dispersive long waves in shallow water. By incorporating fractional calculus operators, the fractional Benney equation provides a more accurate description of wave propagation phenomena in certain physical systems characterized by nonlocal or memory‐dependent behavior. The utilization of the Benney equation enables researchers to simulate these occurrences with greater realism. This study investigates the convergence and inaccuracy of the future scheme. The conformable q‐Shehu homotopy analysis transform method (Cq‐SHATM) generates h‐curves that demonstrate the convergence interval of the series solution obtained. In order to determine the effectiveness and suitability of the Cq‐SHATM, uniqueness and convergence theorems have been proven. This study presents an application that showcases the potential advantages and efficacy of the suggested method. Moreover, an error analysis is conducted to validate the precision of the scheme. Computational simulations are performed to verify the accuracy of the upcoming method. This study presents the results gained from the numerical and graphical analysis. The method presented in this work demonstrates a high level of computational accuracy and simplicity in analyzing and solving complex phenomena associated with conformable fractional nonlinear partial differential equations in the fields of science and technology.

Localized excitation and fractal structures of a (2 + 1)-dimensional Longwater wave equation

Localized excitation and fractal structures of a (2 + 1)-dimensional Longwater wave equation

Oceanic shallow-water investigations on a generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt system

To date, with respect to water waves, researchers have studied certain systems, including a generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt system that describes, e.g., the dispersive long waves in the oceanic shallow water, which we study here. With respect to, e.g., the horizontal velocity of the water wave as well as the height of the deviation from the equilibrium position of the water, with symbolic computation, on the one hand, the system is found to pass the Painlevé test under some coefficient constraints, while on the other hand, two families of the bilinear forms and two families of the N-soliton solutions are constructed, with N as a positive integer. Related constraints are shown. Our bilinear forms and N-soliton solutions depend on the coefficients in the system.

Exploring the new soliton solutions to the nonlinear M-fractional evolution equations in shallow water by three analytical techniques

This paper explores the new soliton solutions of the evolution equations named as truncated M-fractional (1+1)-dimensional non-linear Kaup-Boussinesq system by utilizing the expa function, modified simplest equation and Sardar sub-equation techniques. This system is used in the analysis of long waves in shallow water. The attained results involving trigonometric, hyperbolic and exponential functions. The effect of fractional order derivative is also discussed. Obtained results are very close to the approximate results due to the use of M-fractional derivative. Achieved results are verified by Mathematica tool. Few of the gained results are also explained through 2-D, 3-D and contour graphs. At the end, these techniques are straight forward, useful and effective to deal with non-linear FPDEs.

Exact Solutions of the Stochastic Conformable Broer–Kaup Equations

In this article, the exact solutions of the stochastic conformable Broer–Kaup equations with conformable derivatives which describe the bidirectional propagation of long waves in shallow water are obtained using the modified exponential function method and the generalized Kudryashov method. These exact solutions consist of hyperbolic, trigonometric, rational trigonometric, rational hyperbolic, and rational function solutions, respectively. This shows that the proposed methods are competent and sufficient. In addition, it is aimed to better understand the physical properties by drawing two- and three-dimensional graphics of the exact solutions according to different parameter values. When these exact solutions obtained by two different methods are compared with the solutions attained by other methods, it can be said that these two methods are competent.

A Time Two-Mesh Finite Difference Numerical Scheme for the Symmetric Regularized Long Wave Equation

The symmetric regularized long wave (SRLW) equation is a mathematical model used in many areas of physics; the solution of the SRLW equation can accurately describe the behavior of long waves in shallow water. To approximate the solutions of the equation, a time two-mesh (TT-M) decoupled finite difference numerical scheme is proposed in this paper to improve the efficiency of solving the SRLW equation. Based on the time two-mesh technique and two time-level finite difference method, the proposed scheme can calculate the velocity u(x,t) and density ρ(x,t) in the SRLW equation simultaneously. The linearization process involves a modification similar to the Gauss-Seidel method used for linear systems to improve the accuracy of the calculation to obtain solutions. By using the discrete energy and mathematical induction methods, the convergence results with O(τC2+τF+h2) in the discrete L∞-norm for u(x,t) and in the discrete L2-norm for ρ(x,t) are proved, respectively. The stability of the scheme was also analyzed. Finally, some numerical examples, including error estimate, computational time and preservation of conservation laws, are given to verify the efficiency of the scheme. The numerical results show that the new method preserves conservation laws of four quantities successfully. Furthermore, by comparing with the original two-level nonlinear finite difference scheme, the proposed scheme can save the CPU time.

On a Whitham-Broer-Kaup-like system arising in the oceanic shallow water

Presently, taking care of the oceanic shallow water, people have investigated some Kaup-type systems. For example, a Whitham-Broer-Kaup-like system, which we aim to study hereby, models the dispersive long waves in the oceanic shallow water. As for the horizontal velocity of the water wave and height of the deviation from the equilibrium position of the water, using symbolic computation, we build up (1) a scaling transformation, (2) two hetero-Bäcklund transformations, each of which is from that system to a known linear partial differential equation, and (3) a set of the similarity reductions, from that system to a known ordinary differential equation with some sample solutions also presented. Our results rely on different dispersion/diffusion powers for that system, with respect to the oceanic shallow water.

On a generalized Broer-Kaup-Kupershmidt system for the long waves in shallow water

On a generalized Broer-Kaup-Kupershmidt system for the long waves in shallow water

Interaction structures of multi localized waves within the Kadomtsev–Petviashvili I equation

This paper investigates a (2+1)-dimensional Kadomtsev–Petviashvili I (KP I) equation, which describes the long waves in water with weak surface tension. The KP I equation can be reduced to a (1+1)-dimensional system consisting of a generalized multi component nonlinear Schrödinger (NLS) equation and a modified Korteweg–de Vries (mKdV) equation by using constraints the potential function of the KP I equation to its squared eigenfunction. So, any solution of the (1+1)-dimensional integrable system gives rise to a localized solution of the KP I equation. Then, according to the Lax pairs of the above (1+1)-dimensional system, the generalized (m,N−m)-fold Darboux transformation is adopted to explore various solutions. These solutions include single structure solutions, such as lump-chains, (N-1)-th degenerate lump-chains and Nth-order lump, and especially interaction solutions of arbitrary structures formed on the basis of single structure solutions. As a consequence, the corresponding single structure and mixed interaction solutions for the KP I equation can be obtained by solving the (1+1)-dimensional system with generalized DT. Moreover, all these structure solutions are discussed analytically and shown graphically. Finally, we also sum up various mathematical features of mixed localized wave solutions. The results obtained in this paper may be helpful to understand the interaction phenomena of localized nonlinear waves in KP I equation.

Multiple Equilibria in a Coupled Climate–Carbon Model

Abstract Multiple stable equilibria are intrinsic to many complex dynamical systems, and have been identified in a hierarchy of climate models. Motivated by the idea that the Quaternary glacial–interglacial cycles could have resulted from orbitally forced transitions between multiple stable states mediated by internal feedbacks, this study investigates the existence and mechanisms of multiple equilibria in an idealized, energy-conserving atmosphere–ocean–sea ice general circulation model with a fully coupled carbon cycle. Four stable climates are found for identical insolation and global carbon inventory: an ice-free Warm climate, two intermediate climates (Cold and Waterbelt), and a fully ice-covered Snowball climate. A fifth state, a small ice cap state between Warm and Cold, is found to be barely unstable. Using custom radiative kernels and a thorough sampling of the model’s internal variability, three equilibria are investigated through the state dependence of radiative feedback processes. For fast feedbacks, the systematic decrease in surface albedo feedback from Cold to Warm states is offset by a similar increase in longwave water vapor feedback. At longer time scales, the key role of the carbon cycle is a dramatic lengthening of the adjustment time comparable to orbital forcings near the Warm state. The dynamics of the coupled climate–carbon system are thus not well separated in time from orbital forcings, raising interesting possibilities for nonlinear triggers for large climate changes. Significance Statement How do carbon cycle and other physical processes affect the physical and mathematical properties of the climate system? We use a complex climate model coupled with a carbon cycle to simulate the climate evolution under different initial conditions. Four stable climate states are possible, from the Snowball Earth, in which ice covers the whole planet, to the Warm state, an ice-free world. The carbon cycle drives the global climate change at an extremely slower pace after sea ice retreats. Sea ice and water vapor, on the other hand, constitute the major contributing factors that accelerate faster climate change.

Thinking about the oceanic shallow water via a generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt system

Currently, fluid mechanics has been paid attention to. Hereby, making use of symbolic computation, we investigate a generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt system describing, e.g., the dispersive long waves in the oceanic shallow water. As for, e.g., the horizontal velocity of the water wave and height of the deviation from the equilibrium position of the water, we work out (1) two sets of the hetero-Bäcklund transformations, each of which, from that system to a known linear partial differential equation, and (2) two sets of the similarity reductions, each of which, from that system to a known ordinary differential equation. Our hetero-Bäcklund transformations and similarity reductions depend on the coefficients in that system, as for, e.g., the oceanic shallow water.

A new generalized KdV equation: Its lump-type, complexiton and soliton solutions

A new generalized KdV equation, describing the motions of long waves in shallow water under the gravity field, is considered in this paper. By adopting a series of well-organized methods, the Bäcklund transformation, the bilinear form and diverse wave structures of the governing model are formally extracted. The exact solutions listed in this paper are categorized as lump-type, complexiton, and soliton solutions. To exhibit the physical mechanism of the obtained solutions, several graphical illustrations are given for particular choices of the involved parameters. As a direct consequence, diverse wave structures given in this paper enrich the studies on the KdV-type equations.

Regarding the shallow water in an ocean via a Whitham-Broer-Kaup-like system: hetero-Bäcklund transformations, bilinear forms and [formula omitted] solitons

Considering the water waves, people have investigated many systems. In this paper, what we study is a Whitham-Broer-Kaup-like system for the dispersive long waves in the shallow water in an ocean. With respect to the water-wave horizontal velocity and deviation height from the equilibrium of the water, we construct (A) two branches of the hetero-Bäcklund transformations, from that system to a known constant-coefficient nonlinear dispersive-wave system, (B) two branches of the bilinear forms and (C) two branches of the M-soliton solutions, with M as a positive integer. Results rely upon the oceanic shallow-water coefficients in that system.

Numerical and analytical investigations for solution of fractional Ito equation describing motions of long waves in shallow water under gravity

In this manuscript, a meshfree numerical scheme based on radial basis function-finite difference (RBF-FD) method is proposed to solve the time-fractional Ito equation. First, the temporal derivative is approximated through the finite difference technique. Then, the spatial derivatives are discretized via the multiquadric RBF. Furthermore, some useful theorems are discussed to establish the stability and convergence of the proposed numerical scheme. Finally, some test problems are presented and the results are compared with the acquired Kudryashov method and tanh method solutions to establish the computational efficiency and applicability of the proposed method.

New impressive representations for the soliton behaviors arising from the (2+1)-Boussinesq equation

In the present work, we will detect unpredicted conducts “which weren't realized before” to the soliton solutions of the (2+1)-Boussinesq equation by using three distinct algorithms. The proposed equation is the famous nonlinear one which distinguishes the waves of coastal and ocean engineering that involve the nonlinearity and dispersion terms. Moreover, it is a developed form of the standard Boussinesq equation which describes the solution interaction mechanism of shallow-water waves that involve many waves and shallow water influences refraction, diffraction, shoaling and weak nonlinearity properties arising in fluid dynamics. It also plays principal role in many physics branches, such as propagation of long waves in shallow water, vibrations in a nonlinear string and ion sound waves in plasma and one-dimensional nonlinear lattice waves. The three algorithms that will select for this purpose are the (G′G) -expansion method, the extended direct algebraic method (EDAM) and the extended simple equation method (ESEM). These three distinct manners are applied in the same time and parallel. We will show comparison between our new soliton solution behaviours with that constructed before.