Classical Boussinesq Equation Research Articles

Two-dimensional reductions of the Whitham modulation system for the Kadomtsev–Petviashvili equation

Two-dimensional reductions of the Kadomtsev–Petviashvili(KP)–Whitham system, namely the overdetermined Whitham modulation system for five dependent variables that describe the periodic solutions of the KP equation, are studied and characterized. Three different reductions are considered corresponding to modulations that are independent of x, independent of y, and of t (i.e. stationary), respectively. Each of these reductions still describes dynamic, two-dimensional spatial configurations since the modulated cnoidal wave, generically, has a nonzero speed and a nonzero slope in the xy plane. In all three of these reductions, the integrability of the resulting systems of equations is proven, and various other properties are elucidated. Compatibility with conservation of waves yields a reduction in the number of dependent variables to two, three and four, respectively. As a byproduct of the stationary case, the Whitham modulation system for the classical Boussinesq equation is explicitly obtained.

(G'/G)-EXPANSION METHOD TO SEEK TRAVELING WAVE SOLUTIONS FOR SOME FRACTIONAL NONLINEAR PDES ARISING IN NATURAL SCIENCES

The (G'/G)-expansion method with the aid of symbolic computational system can be used to obtain the traveling wave solutions (hyperbolic, trigonometric and rational solutions) for nonlinear time-fractional evolution equations arising in mathematical physics and biology. In this work, we will process the analytical solutions of the time-fractional classical Boussinesq equation, the time-fractional Murray equation, and the space-time fractional Phi-four equation. With the fact that the method which we will propose in this paper is also a standard, direct and computerized method, the exact solutions for these equations are obtained.

Stability of the 3D Boussinesq equations with partial dissipation near the hydrostatic balance

The Boussinesq equations with partial or fractional dissipation not only naturally generalize the classical Boussinesq equations but also are physically relevant and mathematically important. Unfortunately, it is not often well‐understood for many ranges of fractional powers. This paper focuses on a system of the 3D Boussinesq equations with fractional horizontal and dissipation and proves that if the initial data ( ) in the Sobolev space are close enough to the hydrostatic balance state, respectively, the equations with then always lead to a steady solution.

A numerical experiment on a new piston-type wavemaker: Shallow water approximation

In this paper, we present a numerical experiment on a new piston-type wavemaker using a recently proposed piston-type wavemaker theory. The theory was primarily derived from the classical Boussinesq equation, based on the Pseudo-parameter Iteration Method (PIM). We first use the new theory to observe low amplitude propagating solitary waves and cnoidal waves, whereby we see the workability and validity of the theory. We further succeed in obtaining the graph of a mean power characteristic, in the range of 0.4 ≤kh≤ 1.0, of the new piston-type wavemaker from the new theory.

Analytical solutions for nonlinear systems using Nucci's reduction approach and generalized projective Riccati equations

<abstract><p>In this study, the Nucci's reduction approach and the method of generalized projective Riccati equations (GPREs) were utilized to derive novel analytical solutions for the (1+1)-dimensional classical Boussinesq equations, the generalized reaction Duffing model, and the nonlinear Pochhammer-Chree equation. The nonlinear systems mentioned earlier have been solved using analytical methods, which impose certain limitations on the interaction parameters and the coefficients of the guess solutions. However, in the case of the double sub-equation guess solution, analytic solutions were allowed. The soliton solutions that were obtained through this method display real positive values for the wave phase transformation, which is a novel result in the application of the generalized projective Riccati method. In previous applications of this method, the real positive properties of the solutions were not thoroughly investigated.</p></abstract>

The coupled Boussinesq equation and its Darboux transformation on the time–space scale

Gel’fand-Dikii (GD) formalism is an approach for generating integrable systems in terms of fractional powers of the δ differential operator. In this paper, it extends the GD formalism associated with the third-order δ differential operator L to the time scale. Then, the coupled Boussinesq equation on the time–space scale is given by taking special values, and it can be reduced on different time–space scales. Moreover, the exact solutions of the coupled Boussinesq equation on the time–space scale and the classical Boussinesq equation are constructed via employing the extensions of the Darboux theorem and Crum theorem on the time scale.

General Rational Solutions In the Classical Boussinesq Equation

General Rational Solutions In the Classical Boussinesq Equation

New nonlinear theory for a piston-type wavemaker: The classical Boussinesq equations

In this study, we present a new nonlinear theory for a moving boundary wavemaker of piston-type based on a nonlinear dispersive shallow water model, where the classical Boussinesq equations are employed as a starting point. The new theory is inherently different from the traditional wavemaker theories, such as the usual theories employed for solving the Laplace equation equipped with the free surface boundary conditions by using the perturbation approach. To verify the wavemaker theory proposed in this study, the ratio of the wave height relative to the stroke characterizing the performance of the wavemaker was observed and compared with numerical, experimental, and Havelock's theoretical results, thereby confirming that the results obtained with the proposed theory were in significant agreement. Furthermore, a comparison of the solitary wave generated by the proposed theory and the known exact solution showed that they were in good agreement.

Exact solution of Boussinesq equations for propagation of nonlinear waves

In this paper, we consider two Boussinesq models that describe propagation of small-amplitude long water waves. Exact solutions of the classical Boussinesq equation that represent the interaction of wave packets and waves on solitons are found. We use the Hirota representation and computer algebra methods. Moreover, we find various solutions for one of the variants of the Boussinesq system. In particular, these solutions can be interpreted as the fusion and decay of solitary waves, as well as the interaction of more complex structures.

Asymptotic dynamics for the small data weakly dispersive one-dimensional Hamiltonian ABCD system

Consider the Hamiltonian a b c d abcd system in one dimension, with data posed in the energy space H 1 × H 1 H^1\times H^1 . This model, introduced by Bona, Chen, and Saut, is a well-known physical generalization of the classical Boussinesq equations. The Hamiltonian case corresponds to the regime where a , c > 0 a,c>0 and b = d > 0 b=d>0 . Under this regime, small solutions in the energy space are globally defined. A first proof of decay for this 2 × 2 2\times 2 system was given in [J. Math. Pure Appl. (9) 127 (2019), 121–159] in a strongly dispersive regime, i.e., under essentially the conditions \[ b = d > 2 9 , a , c > − 1 18 . b=d > \frac 29, \quad a,c>-\frac 1{18}. \] Additionally, decay was obtained inside a proper subset of the light cone ( − | t | , | t | ) (-|t|,|t|) . In this paper, we improve [J. Math. Pure Appl. (9) 127 (2019), 121–159] in three directions. First, we enlarge the set of parameters ( a , b , c , d ) (a,b,c,d) for which decay to zero is the only available option, considering now the so-called weakly dispersive regime a , c ∼ 0 a,c\sim 0 : we prove decay if now \[ b = d > 3 16 , a , c > − 1 48 . b=d > \frac 3{16}, \quad a,c>-\frac 1{48}. \] This result is sharp in the case where a = c a=c , since for a , c a,c bigger, some a b c d abcd linear waves of nonzero frequency do have zero group velocity. Second, we sharply enlarge the interval of decay to consider the whole light cone, that is to say, any interval of the form | x | ∼ | v | t |x|\sim |v|t for any | v | > 1 |v|>1 . This result rules out, among other things, the existence of nonzero speed solitary waves in the regime where decay is present. Finally, we prove decay to zero of small a b c d abcd solutions in exterior regions | x | ≫ | t | |x|\gg |t| , also discarding super-luminical small solitary waves. These three results are obtained by performing new improved virial estimates for which better decay properties are deduced.

Waves and Structures in the Boussinesq Equations

The classical Boussinesq equation describing gravity waves in shallow waters is under consideration. Hirota’s bilinear representation is used to construct exact solutions describing wave packets, waves on solitons, and “dancing” waves. The principle of multiplying the solutions of the Hirota equation is formulated, which helps constructing more complex structures made of solitons, wave packets, and other types of waves.

On the accurate simulation of nearshore and dam break problems involving dispersive breaking waves

The ability of numerical models to deal with wave breaking processes and dry areas is of paramount importance for applications in coastal zones and dam breaks. The mathematical models commonly used in such real problems are usually based on Boussinesq-type equations and, to a small extend, on Serre equations. However, these standard models are weakly dispersive and must be appropriately modified to deal with breaking waves and dry areas. Indeed, nearshore and dam break problems involve complex wave dynamics and highly dispersive wave processes can easily arise. In those cases, it is well known that weakly dispersive models like the ones based on classical Boussinesq or Serre equations are unreliable for an accurate simulation of the phenomena involved.In this work we extend the applicability of an improved Serre model, herein denoted by Serreα,β, to include the wave breaking process, the broken waves propagation, and dry areas. We provide a comprehensive set of numerical examples involving wave propagation over exposed and submerged structures, as well as dam break problems. The numerical experiments show the accuracy and robustness of the proposed model. Particular attention is given to bottom friction modeling, where the standard Manning’s assumption is compared with a more realistic formulation. Also noteworthy is the simulation of wave breaking problems with highly dispersive effects. The advantages of the Serreα,β model over the standard Serre model for these challenging cases are clear.

Long wave approximation for water waves under a Coriolis forcing and the Ostrovsky equation

This paper is devoted to the study of the long wave approximation for water waves under the influence of the gravity and a Coriolis forcing. We start by deriving a generalization of the Boussinesq equations in one (spatial) dimension and we rigorously justify them as an asymptotic model of water wave equations. These new Boussinesq equations are not the classical Boussinesq equations: a new term due to the vorticity and the Coriolis forcing appears that cannot be neglected. We study the Boussinesq regime and derive and fully justify different asymptotic models when the bottom is flat: a linear equation linked to the Klein–Gordon equation admitting the so-called Poincaré waves; the Ostrovsky equation, which is a generalization of the Korteweg–de Vries (KdV) equation in the presence of a Coriolis forcing, when the rotation is weak; and the KdV equation when the rotation is very weak. Therefore, this work provides the first mathematical justification of the Ostrovsky equation. Finally, we derive a generalization of the Green–Naghdi equations in one spatial dimension for small topography variations and we show that this model is consistent with the water wave equations.

An improvement of convergence of a dispersion-relation preserving method for the classical Boussinesq equation

A dispersion-relation preserving (DRP) method, as a semi-analytic iterative procedure, has been proposed by Jang (2017) for integrating the classical Boussinesq equation. It has been shown to be a powerful numerical procedure for simulating a nonlinear dispersive wave system because it preserves the dispersion-relation, however, there still exists a potential flaw, e.g., a restriction on nonlinear wave amplitude and a small region of convergence (ROC) and so on. To remedy the flaw, a new DRP method is proposed in this paper, aimed at improving convergence performance. The improved method is proved to have convergence properties and dispersion-relation preserving nature for small waves; of course, unique existence of the solutions is also proved. In addition, by a numerical experiment, the method is confirmed to be good at observing nonlinear wave phenomena such as moving solitary waves and their binary collision with different wave amplitudes. Especially, it presents a ROC (much) wider than that of the previous method by Jang (2017). Moreover, it gives the numerical simulation of a high (or large-amplitude) nonlinear dispersive wave. In fact, it is demonstrated to simulate a large-amplitude solitary wave and the collision of two solitary waves with large-amplitudes that we have failed to simulate with the previous method. Conclusively, it is worth noting that better convergence results are achieved compared to Jang (2017); i.e., they represent a major improvement in practice over the previous method.

Some applications of the (G′/G, 1/G)-expansion method to find new exact solutions of NLEEs

The double ( $G'/G$ , $1/G$ )-expansion method is an influential, effective and well-suited method to examine closed form traveling wave solutions to nonlinear evolution equations (NLEEs). In this article, we extract abundant wave solutions to the (2+1)-dimensional typical breaking soliton equation and the (1+1)-dimensional classical Boussinesq equation through this method. The wave solutions are presented in terms of hyperbolic function, trigonometric function and rational function. By means of the wave transformation, the NLEEs are reduced to nonlinear ordinary differential equation (ODE) and then the nonlinear ODE is utilized to examine the necessary NLEE. The method can be considered as the generalization of the ( $G'/G$ -expansion method established by Wang et al. and it is shown that the suggested method is a powerful mathematical tool for investigating nonlinear evolution equations.

A new dispersion-relation preserving method for integrating the classical Boussinesq equation

In this paper, a dispersion-relation preserving method is proposed for nonlinear dispersive waves, starting from the oldest weakly nonlinear dispersive wave mathematical model in shallow water waves, i.e., the classical Boussinesq equation. It is a semi-analytic procedure, however, which preserves, as a distinctive feature, the dispersion-relation imbedded in the model equation without adding (unwelcome) numerical effects, i.e., the proposed method has the same dispersion-relation as the original classical Boussinesq equation. This remarkable (dispersion-relation) preserving property is proved mathematically for small wave motion in present study. The property is also numerically examined by observing both the local wave number and the local frequency of a slowly varying water-wave group. The dispersion-relation preserving method proposed here is powerful as well for observing nonlinear wave phenomena such as solitary waves and their collision. In fact, the main features of nonlinear wave characteristics are clearly seen through not only a single propagating solitary wave but counter-propagating (head-on) solitary wave collisions. They are compared with known (exact) nonlinear solutions, the results of which represent a major improvement over existing solution formulations in the literature.

On the study of a nonlinear higher order dispersive wave equation: its mathematical physical structure and anomaly soliton phenomena

This paper introduces a systematic approach to investigate a higher order nonlinear dispersive wave equation for modeling different wave modes. We present both the conventional KdV-type soliton and anomaly type solitons for the equation. We also show the conservation laws and Hamiltonian structures for the equation. Our results suggest that the underlying equation has more interacting soliton phenomena than one would have known for the classical KdV and Boussinesq equation.

The classical Korteweg capillarity system: geometry and invariant transformations

A class of invariant transformations is presented for the classical Korteweg capillarity system. The invariance is an extension of a kind originally introduced in an anisentropic gasdynamics context. In a particular instance, application of the invariant transformation leads to a deformed one-parameter class of Kármán–Tsien-type capillarity laws associated with a deformation of an integrable nonlinear Schrödinger-type equation which incorporates a de Broglie–Bohm potential. The latter and another integrable case associated with the classical Boussinesq equation may be linked to the motion of curves in Euclidean and projective space so that both the invariant transformation and the Galilean invariance of the capillarity system may be interpreted in a geometric and soliton-theoretic manner. The work is set in the broader context of other connections of invariant transformations in gasdynamics with soliton theory.

The exp(−Φ(η))-expansion method with application in the (1+1)-dimensional classical Boussinesq equations

Periodic and soliton solutions are presented for the (1+1)-dimensional classical Boussinesq equation which governs the evolution of nonlinear dispersive long gravity wave traveling in two horizontal directions on shallow water of uniform depth. The equation is handled via the exp(−Φ(η))-expansion method. It is worth declaring that the method is more effective and useful for solving the nonlinear evolution equations. In particular, mathematical analysis and numerical graph are provided for those solitons, periodic, singular kink and bell type solitary wave solutions to visualize the dynamics of the equation.

Boussinesq and Serre type models with improved linear dispersion characteristics: Applications

The classical Boussinesq equations only incorporate weak dispersion and weak nonlinearity, and are valid only for long waves in shallow waters. The classical Serre equations (or Green and Naghdi) are fully-nonlinear and weakly dispersive. Thus, as for the classical Boussinesq models, Serre's equations are valid only for shallow water conditions. To allow applications in a greater range of depth to wavelength ratio, a new set of extended Serre equations with additional terms of dispersive origin is proposed in this work. The equations are solved using an efficient finite-difference method, which consistency and stability are tested by comparison with a closed-form solitary wave solution of these equations. It is shown that computed results agree closely with the analytical ones and test data. An equivalent form of the Boussinesq equations, also with improved linear dispersion characteristics, is solved using a numerical procedure similar to that used to solve the extended Serre equations. Numerical solutions of both approaches, in the case of waves generated by disturbances on the water surface, are compared to each other and with available data, testing different functions commonly used in modelling the generation and propagation of ship waves.