Shallow Water Wave Model Research Articles

Complex Wave Surfaces to the Extended Shallow Water Wave Model with (2+1)-dimensional

In this paper, we apply an analytical method, namely, the sine-Gordon expansion method and extract some complex optical soliton solutions to the (2+1)-dimensional extended shallow water wave model, which describes the evolution of shallow water wave propagation. We obtain some complex mixed-dark and bright soliton solutions to this nonlinear model. Considering some suitable values of parameters, we plot the various dimensional simulations of every results found in this manuscript. We observe that our result may be useful in detecting some complex waves behaviors.

Formula omitted]-lump and hybrid solutions of a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation

In this paper, the N-soliton solutions of a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation are obtained by means of the bilinear method. By applying the long wave limit to the N-solitons, the M-lump waves are constructed. The propagation orbits, velocities and the collisions among the lumps of the M-lump waves are analyzed. Three kinds of high-order hybrid solutions are presented, which contain the hybrid solution between lumps and solitons, a 1-lump and 1-breather, and a m-breather and n-soliton. The results are helpful to explain some nonlinear phenomena of the generalized shallow water wave model.

Exact Traveling and Nano-Solitons Wave Solitons of the Ionic Waves Propagating along Microtubules in Living Cells

In this paper, the weakly nonlinear shallow-water wave model is mathematically investigated by applying the modified Riccati-expansion method and Adomian decomposition method. This model is used to describe the propagation of waves in weakly nonlinear and dispersive media. We obtain exact and solitary wave solutions of this model by using the modified Riccati-expansion method then using these solutions to determine the boundary and initial conditions. These conditions are employed to evaluate the semi-analytical wave solutions and calculate the absolute value of error. The values of absolute error show the accuracy of the obtained solutions. Some solutions are sketched to show the perspective view of the solution of this model. Moreover, the novelty of the obtained solutions is illustrated by showing the similarity and differences between our and previous solutions of the model.

Completing the Study of Traveling Wave Solutions for Three Two-Component Shallow Water Wave Models

For three two-component shallow water wave models, from the approach of dynamical systems and the singular traveling wave theory developed in [Li & Chen, 2007], under different parameter conditions, all possible bounded solutions (solitary wave solutions, pseudo-peakons, periodic peakons, as well as smooth periodic wave solutions) are derived. More than 19 explicit exact parametric representations are obtained. Of more interest is that, for the integrable two-component generalization of the Camassa–Holm equation, it is found that its [Formula: see text]-traveling wave system has a family of pseudo-peakon wave solutions. In addition, its [Formula: see text]-traveling wave system has two families of uncountably infinitely many solitary wave solutions. The new results complete a recent study by Dutykh and Ionescu-Kruse [2016].

Far-Field Characteristics of Linear Water Waves Generated by a Submerged Landslide over a Flat Seabed

Understanding the propagation of landslide-generated water waves is of great help against tsunami hazards. In order to investigate the effects of landslide shapes on the far-field leading wave generated by a submerged landslide at a constant depth, three linear wave models with different degrees of dispersive properties are employed in this study. The linear fully dispersive model is then validated by comparing the results against the experimental data available for landslides with a low Froude number. Three simplified shapes of landslides with the same volume, which are unnatural for a body of incoherent material, are used to investigate the effects of landslide shapes on the far-field properties of the generated leading wave over a flat seabed. The results show that the far-field leading crest over a constant depth is independent of the exact landslide shape and is invalid at a shallow water depth. Therefore, the most popular non-dispersive model (also called the shallow water wave model) cannot be used to reproduce the phenomenon. The weakly dispersive wave model can predict this phenomenon well. If only the leading wave is considered, this model is accurate up to at least μ = h0/Lc = 0.6, where h0 is the water depth and Lc denotes the characteristic length of the landslide.

Decay in the one dimensional generalized Improved Boussinesq equation

We consider the decay problem for the generalized improved (or regularized) Boussinesq model with power type nonlinearity, a modification of the originally ill-posed shallow water waves model derived by Boussinesq. This equation has been extensively studied in the literature, describing plenty of interesting behavior, such as global existence in the space $$H^1\times H^2$$, existence of super luminal solitons, and lack of a standard stability method to describe perturbations of solitons. The associated decay problem has been studied by Liu, and more recently by Cho–Ozawa, showing scattering in weighted spaces provided the power of the nonlinearity p is sufficiently large. In this paper we remove that condition on the power p and prove decay to zero in terms of the energy space norm $$L^2\times H^1$$, for any $$p>1$$, in two almost complementary regimes: (1) outside the light cone for all small, bounded in time $$H^1\times H^2$$ solutions, and (2) decay on compact sets of arbitrarily large bounded in time $$H^1\times H^2$$ solutions. The proof consists in finding two new virial type estimates, one for the exterior cone problem based in the energy of the solution, and a more subtle virial identity for the interior cone problem, based in a modification of the momentum.

The solitary wave solutions of the nonlinear perturbed shallow water wave model

In this paper, we study the nonlinear perturbed shallow water wave model, which satisfies the asymptotic integrability condition with this family: the KdV equation, Camassa–Holm equation and Degasperis–Procesi equation. We establish the existence of solitary wave solutions for the perturbed nonlinear dispersive equations by means of the geometric singular perturbation and invariant manifold theory.

New Analytical Solutions of Conformable Time Fractional Bad and Good Modified Boussinesq Equations

Abstract The main purpose of this article is to obtain the new solutions of fractional bad and good modified Boussinesq equations with the aid of auxiliary equation method, which can be considered as a model of shallow water waves. By using the conformable wave transform and chain rule, nonlinear fractional partial differential equations are converted into nonlinear ordinary differential equations. This is an important impact because both Caputo definition and Riemann–Liouville definition do not satisfy the chain rule. By using conformable fractional derivatives, reliable solutions can be achieved for conformable fractional partial differential equations.

New dark-bright soliton in the shallow water wave model

In this paper, we employ the sine-Gordon expansion method to shallow water wave models which are Kadomtsev-Petviashvili-Benjamin-Bona-Mahony and the Benney-Luke equations. We construct many new complex combined dark-bright soliton, anti-kink soliton solutions for the governing models. The 2D, 3D and contour plots are given under the suitable coefficients. The obtained results show that the approach proposed for these completely integrable equations can be used effectively.

On local-in-space blow-up scenarios for a weakly dissipative rotation-Camassa–Holm equation

ABSTRACT In this paper, we mainly devote to investigate a weakly dissipative shallow water wave model. By overcoming the difficulties caused by the complicated higher-order nonlinear terms and dissipative term, we present blow-up scenarios with dissipative parameter in the non-periodic case under different initial conditions. Meanwhile, we obtain the blow-up result with in the periodic case.

BIFURCATIONS AND EXACT TRAVELLING WAVE SOLUTIONS FOR A SHALLOW WATER WAVE MODEL WITH A NON-STATIONARY BOTTOM SURFACE

We consider a shallow water wave model with a non-stationary bottom surface. By applying dynamical system approach to the model problem, we are able to obtain all possible bounded solutions (compactons, solitary wave solutions and periodic wave solutions) under different parameter conditions. More than 19 exact parametric representations are provided explicitly.

Energy accumulation during the growth of forced wave induced by a moving atmospheric pressure disturbance

ABSTRACTAtmospheric pressure disturbances of high translational speed can excite significant water waves, which are widely believed to be the main cause of destructive meteotsunamis. During the response of sea surface to adisturbance, energy is transferred from the atmosphere to the ocean, while the forced wave induced by the disturbance gradually grows and finally reaches a quasi-steady state. This study aims at quantifying the time required to excite the forced wave. Based on the nonlinear shallow water wave model, the evolution of the forced wave is numerically investigated. Analyses of the accumulation of wave energy are imported to help understand the evolution of the forced wave in different situations. The characteristics of energy accumulation show a great difference in the cases of different Froude number Fr. A parameter named response time is defined to quantify the time required for the evolution of the forced wave. It can be seen that Fr is the principal parameter that controls the wave evolution, while the effects of nonlinearity and bottom friction are significant only when Fr is close to 1. Simple relationships are proposed to roughly estimate the response time for Fr<0.9 and Fr>1.1, which can also provide a lower limit for Fr≈1.

Construction of solitary wave solutions of some nonlinear dynamical system arising in nonlinear water wave models

The higher order of nonlinear partial differential equations in mathematical physics is studied. We used the analytical mathematical methods of the nonlinear (3+1)-dimensional extended Zakharov–Kuznetsov dynamical, modified KdV–Zakharov–Kuznetsov and generalized shallow water wave equations to demonstrate the efficiency and validity of the proposed powerful technique. The shallow water wave models have been applied in tidal waves and weather simulation. Exact wave solutions of these models in various forms such as Kink and anti-Kink solitons, bright–dark soliton, solitary wave and periodic solutions are constructed that have plenty of applications in diverse areas of physics. Graphically, we presented the movement of some obtained solitary wave solutions that aids in understanding the physical phenomena of these models.

New Wave Solutions of Time-Fractional Coupled Boussinesq–Whitham–Broer–Kaup Equation as A Model of Water Waves

The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.

Well-posedness and wave breaking for a shallow water wave model with large amplitude

Considered herein is the Cauchy problem of a model for shallow water waves of large amplitude. Using Littlewood–Paley decomposition and transport equation theory, we establish the local well-posedness of the equation in Besov spaces $$B^s_{p,r} $$ with $$1\le p,r \le +\infty $$ and $$s>\max \{1+\frac{1}{p },\frac{3}{2}\}$$ (and also in Sobolev spaces $$H^s=B^s_{2,2}$$ with $$s>3/2$$). Then, the precise blow-up mechanism for the strong solutions is determined in the lowest Sobolev space $$H^s $$ with $$s>3/2$$. Our results improve the corresponding work for this model in Quirchmayr (J Evol Equ 16:539–567, 2016), in which the Sobolev index $$s=3$$ is required. In addition, we also investigate the asymptotic behaviors of the strong solutions to this equation at infinity within its lifespan provided the initial data lie in weighted $$L_{p,\phi }:=L_p(\mathbb {R},\phi ^pdx)$$ spaces.

The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise

This paper is concerned with the Camassa-Holm-KP equation, which is a model for shallow water waves. By using the analysis of the phase space, we obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the Camassa-Holm-KP equation without delay. Furthermore we show the existence of solitary wave solutions for the equation with a special local delay convolution kernel by combining the geometric singular perturbation theory and invariant manifold theory. In addition, we discuss the existence of solitary wave solutions for the Camassa-Holm-KP equation with strength $ 1 $ of nonlinearity, and prove the monotonicity of the wave speed by analyzing the ratio of the Abelian integral.

More on Bifurcations and Dynamics of Traveling Wave Solutions for a Higher-Order Shallow Water Wave Equation

This paper studies the dynamics of traveling wave solutions to a shallow water wave model with a large-amplitude regime in phase space. The corresponding traveling wave system is a singular planar dynamical system with two singular straight lines. By using the method of dynamical systems, bifurcation diagrams are obtained. The existence of solitary wave solutions, periodic wave solutions, peakon, pseudo-peakon solution, periodic peakon solutions and compacton solutions are determined under different parameter conditions.

Dynamics of mixed lump-solitary waves of an extended (2 + 1)-dimensional shallow water wave model

To explore the features of lump solutions, which are local in every direction of space, a (2+1)-dimensional extended shallow water wave model is studied, based on its bilinear representation. Several ansatzes have been utilized to determine single lump waves, lump-kink waves, single kinks and multi-lumps leading to breathers in terms of function patterns for the model. Through analyzing interactions between solitons, the impact of free parameters involved in the solutions on interaction types is exhibited. We determine a condition on the parameters under which a single kink wave can be converted into a multi-lump wave. To illustrate the interaction of exponential and periodic function waves, we show that multi-lump waves in the form of breather waves especially come into sight as a straight line or an X shape. To realize dynamics, we make various graphical analyses on the presented solutions, which gives an essential improvement in the physical realizing of higher-dimensional lump waves in oceanography and nonlinear optics.

Finite Difference Methods for Simulation of Water Waves Generated by Moving Topography

In this paper, we consider water waves excited by moving topography. This problem usually occurs in the tsunami generation and propagation. We use the shallow water wave model governing this wave generation and propagation problem. Two finite difference methods are tested to solve the problem. The first is the forward in time centred in space finite difference method. The second is the Adams-Bashforth--Adams-Moulton method. We obtain that the second gives more flexibility in terms of the choice of time step value.

The existence of solitary wave solutions of delayed Camassa–Holm equation via a geometric approach

This paper is concerned with the Camassa–Holm equation, which is a model for shallow water waves. We first establish the existence of solitary wave solutions for the equation without delay. And then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system, especially the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Camassa–Holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equation with disturbance also possesses homoclinic orbit, and there exists solitary wave solution of the delayed Camassa–Holm equation.