Modified Boussinesq Equation Research Articles

Discrete modified Boussinesq equation and Darboux transformation

In this letter, by a 3-periodic reduction technique, we regain the well-known discrete modified Boussinesq equation and its Lax pairs. It is the first time that the Darboux transformation is successfully constructed for this discrete equation. We then obtain its explicit solutions in terms of Casoratian determinants. The derived solutions are soliton solutions when the seed solutions are taken as v=1 and w=1.

Effects of Unsaturated Flow on Hillslope Recession Characteristics

AbstractRecession flow analysis is usually conducted to infer hydraulic parameters of hillslope aquifers. Various Boussinesq equation‐based models, both linear and nonlinear, have been used to analyze the recession curves for sloping aquifers, with a focus on the long‐time recession behavior. Based on a modified Boussinesq equation with capillarity incorporated, we demonstrate the significant effect of unsaturated flow on the recession curve, which result in three (instead of two) power law regimes with two transition points (instead of one) corresponding to the formation of a fully unsaturated zone at the adjacent area of the upslope boundary and across the whole domain, respectively. The results show that the power of the second and third recession regime is variable, depending on the slope angles, soil types, and hillslope geometries. The unsaturated flow effects also lead to the absence of drastic drop of at the transition between the first and second regime, which was predicted by previous numerical models but has not been observed in the field or laboratory experiments. These findings have important implications for recession flow analysis in studies of hillslope aquifers.

Boussinesq modeling of solitary wave run-up reduction by emergent vegetation on a sloping beach

To better understand the reduction of the run-up of leading tsunami waves by mangrove forests on a sloping beach, we presented a numerical model in this study, which was based on one-dimensional (1D) fully nonlinear Boussinesq equations with the effect of vegetation resistance included in terms of the drag coefficient. Solitary waves were used to model the leading tsunami waves and five models of different forest densities and tree distributions of the emergent vegetation were examined. The bottom friction and vegetation drag were calibrated based on our published laboratory experiments. Model results show that the modified Boussinesq equations could give satisfactory predictions of the solitary wave transformation over the vegetation and subsequent run-up on the slope. The drag coefficient was found to be dependent on the incident wave height. Effects of beach slope, forest density and tree distribution on the drag coefficient were also discussed based on the numerical results.

Asymptotics for the modified Boussinesq equation in one space dimension

We consider the Cauchy problem for the modified Boussinesq equation in one space dimension \begin{equation*} \begin {cases} w_{tt}=a^{2}\partial _{x}^{2}w-\partial _{x}^{4}w+\partial _{x}^{2} ( w^{3} ) ,\text{ } ( t,x ) \in \mathbb{R}^{2}, \\ w ( 0,x ) =w_{0} ( x ) ,\text{ }w_{t} ( 0,x ) =w_{1} ( x ) ,\text{ }x\in \mathbb{R}\text{,} \end {cases} \end{equation*} where $a > 0.$ We study the large time asymptotics of solutions to the Cauchy problem for the modified Boussinesq equation. We apply the factorization technique developed recently in papers [5], [6], [7], [8].

Finite Difference Method for Modified Boussinesq Equation

In this paper a numerical of the solution of the Cauchy problem for the nonlinear modified Boussinesq equation (or IMBq equation) is studied. This equation with the boundary conditions models the propagation of waves in shallow water, taking into account capillary effects and the preservation of mass in the layer, filtration of water in the soil, as well as shock waves. In the case when the equation is nondegenerate, a global solution and a solution in the form of solitons are obtained. In the degenerate case, the existence of a unique local solution was proved by the methods of phase space and the theory of relatively limited ones developed by G. Sviridyuk and his students, as well as the theory of differentiable Banach manifolds. A numerical study of this problem by the modified Galerkin method has already been carried out earlier. However, the operation time of algorithms based on modified Galerkin method it rapidly increases with an increase amount of Galerkin sum. In this article, a numerical study is carried out by the finite difference method. The Cauchy -- Dirichlet problem for the IMBq equation is reduced to an implicit difference problem. A comparison is made of the speed of the modified Galerkin method and the finite difference method.

Generalized Wronskian solutions of modified Boussinesq equation

The generalized Wronskian solutions whose elements satisfy matrix equation of linear problem for the modified Boussinesq equation are obtained through Wronskian technique. Furthermore, rational solutions, Matveev solutions, complexiton solutions and interaction solutions are derived by taking special cases in general solutions.

On the Strong Solutions and the Structural Stability of the g-Bénard Problem

ABSTRACTIn this paper, we study a type of modified Boussinesq equations which is called g-Bénard problem. We show the existence and uniqueness of strong solutions of the problem in two dimensions, and then, we investigate the continuous dependence of the solutions on the viscosity parameter.

New exact solutions for the Wick-type stochastic modified Boussinesq equation for describing wave propagation in nonlinear dispersive systems

In this article, exact solutions of a Wick-type stochastic fractional modified Boussinesq equation have been obtained by using an improved sub-equation method. We have used the Hermite transform for transforming the Wick-type stochastic modified Boussinesq equation to a deterministic partial differential equation. Also we have applied the inverse Hermite transform for obtaining a set of stochastic solutions in the white noise space.

Formula: see text] graded discrete Lax pairs and Yang-Baxter maps.

We recently introduced a class of [Formula: see text] graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In this paper, we introduce the corresponding Yang-Baxter maps. Many well-known examples belong to this scheme for N=2, so, for N≥3, our systems may be regarded as generalizations of these. In particular, for each N we introduce a class of multi-component Yang-Baxter maps, which include HBIII (of Papageorgiou et al. 2010 SIGMA 6, 003 (9 p). (doi:10.3842/SIGMA.2010.033)), when N=2, and that associated with the discrete modified Boussinesq equation, for N=3. For N≥5 we introduce a new family of Yang-Baxter maps, which have no lower dimensional analogue. We also present new multi-component versions of the Yang-Baxter maps FIV and FV (given in the classification of Adler et al. 2004 Commun. Anal. Geom. 12, 967-1007. (doi:10.4310/CAG.2004.v12.n5.a1)).

ANALYTICAL SOLITARY WAVE SOLUTIONS FOR THE NONLINEAR ANALOGUES OF THE BOUSSINESQ AND SIXTH-ORDER MODIFIED BOUSSINESQ EQUATIONS

Using tanh function and polynomial function methods, analytical solitary wave solutions have been found for the nonlinear analogues of Boussinesq and sixth-order modified Boussinesq equations where the nonlinearity is in the time-derivative term.

On properties of solutions to the improved modified Boussinesq equation

On properties of solutions to the improved modified Boussinesq equation

Periodic wave solutions and solitary wave solutions of generalized modified Boussinesq equation and evolution relationship between both solutions

In this paper, we focus on studying the exact solitary wave solutions and periodic wave solutions of the generalized modified Boussinesq equation utt−δuttxx−(a1u+a2u2+a3u3)xx=0, as well as the evolution relationship between these solutions. Detailed qualitative analysis is conducted on traveling wave solutions of this equation, and global phase portraits in various parameter conditions are proposed. Various significant results about the existence of both solutions, including three forms of solitary wave solutions and four exact bounded periodic wave solutions in different conditions are obtained. Then, we further discuss the relationship between energy of Hamiltonian system corresponding to this equation and the periodic wave solutions and solitary wave solutions. It is concluded that the essential reason of periodic wave solutions and solitary wave solutions is the different values for the energy of Hamiltonian system corresponding to this equation. In addition, the limited relations of periodic wave solutions and solitary wave solutions with the energy of Hamiltonian system are proposed, and the schematic diagram of evolution from periodic wave solutions to solitary wave solutions is drawn.

CTE Solvability, Nonlocal Symmetry and Explicit Solutions of Modified Boussinesq System**Supported by the National Natural Science Foundation of China under Grant Nos. 11305106 and 11505154

A consistent tanh expansion (CTE) method is used to study the modified Boussinesq equation. It is proved that the modified Boussinesq equation is CTE solvable. The soliton-cnoidal periodic wave is explicitly given by a nonanto-BT theorem. Furthermore, the nonlocal symmetry for the modified Boussinesq equation is obtained by the Painlevé analysis. The nonlocal symmetry is localized to the Lie point symmetry by introducing one auxiliary dependent variable. The finite symmetry transformation related with the nonlocal symemtry is obtained by solving the initial value problem of the prolonged systems. Thanks to the localization process, many interaction solutions among solitons and other complicated waves are computed through similarity reductions. Some special concrete soliton-cnoidal wave interaction behaviors are studied both in analytical and graphical ways.

CRE Solvability, Exact Soliton-Cnoidal Wave Interaction Solutions, and Nonlocal Symmetry for the Modified Boussinesq Equation

It is proved that the modified Boussinesq equation is consistent Riccati expansion (CRE) solvable; two types of special soliton-cnoidal wave interaction solution of the equation are explicitly given, which is difficult to be found by other traditional methods. Moreover, the nonlocal symmetry related to the consistent tanh expansion (CTE) and the residual symmetry from the truncated Painlevé expansion, as well as the relationship between them, are obtained. The residual symmetry is localized after embedding the original system in an enlarged one. The symmetry group transformation of the enlarged system is derived by applying the Lie point symmetry approach.

The motion of a thin liquid layer on the outer surface of a rotating cylinder

We derive the shallow water equations describing the motion of a thin liquid film on the outer surface of a rotating cylinder. These equations are an analogue of the modified Boussinesq equations describing shallow water flows with constant vorticity. The standard multi-scale methods are employed to construct asymptotic equations in the long-wave approximation. These asymptotic equations are analyzed using the hodograph method. It is found that for the particular case of a dispersionless irrotational flow, the equations describing flows on the outer surface of a cylinder reduce to elliptic equations. Numerical evaluation of the exact solutions obtained shows that the asymptotic equations possess a rich variety of solutions representing various wave patterns.

Modeling of Time-Varying Shear Current Field Under Breaking and Broken Waves with Surface Rollers

This study aims to develop a new numerical model for predictions of the time-varying shear current field under breaking and broken waves. The present model differs from the other similar existing models in that the model explicitly determines the bottom shear stress so that it satisfies both force balance and mass-conservation equations while most of the other existing models leave the bottom shear stress as one of the calibration parameters to be fitted. The present model consists of two sub-models, a wave model and a flow model. The wave model is based on modified Boussinesq equations with evolution and dissipation of surface rollers. The flow model is based on Reynolds' equations with turbulence closure models and computes vertical profiles of wave-induced shear current field. The numerical model is applied to various existing and newly performed experimental cases which cover various breaker types and bed profiles with different bed roughness. The present model showed overall good predictive skills of various surf zone hydrodynamics such as wave heights, mean water levels and undertow profiles. It should also be highlighted that the model has only a few calibration parameters and the predictive skill of the model is relatively less sensitive to these parameters.

Group classification and exact solutions of generalized modified Boussinesq equation

We present symmetry classification and exact solutions of generalized modified Boussinesq (GMB) equation. The direct method of group classification is utilized to determine four different functional forms of . The GMB equation admits two-dimensional principle algebra for arbitrary and the algebra extends to three-dimensional for other forms of . Similarity reductions are made in each case and exact solutions are derived.

The Propagation of Shock Waves in Incompressible Fluids: The Case of Freshwater

In this paper we investigate the basic features of shock waves propagation in freshwater in the framework of a hyperbolic model consisting of the one-dimensional Euler equations closed by means of polynomial equations of state extracted from experimental tabulated data available in the literature (Sun et al. in Deep-Sea Res. I 55:1304---1310, 2008). The Rankine---Hugoniot equations are numerically solved in order to determine the Hugoniot locus representing the set of perturbed states that can be connected through a k-shock to an unperturbed state. The results are found to be consistent with those previously obtained in the framework of the EQTI model by means of a modified Boussinesq equation of state.

Application of the bifurcation method to the modified Boussinesq equation

In this paper, we investigate the modified Boussinesq equation utt uxx # uxxxx 3(u 2 )xx + 3(u 2 ux)x = 0. Firstly, we give a property of the solutions of the equation, that is, if 1 + u(x,t) is a solution, so is 1 u(x,t). Secondly, by using the bifurcation method of dynamical sys- tems we obtain some explicit expressions of solutions for the equation, which include kink-shaped solutions, blow-up solutions, periodic blow-up solutions and solitary wave solutions. Some previous results are extended.

Non-Commutative Rational Yang–Baxter Maps

Starting from multidimensional consistency of non-commutative lattice modified Gel'fand-Dikii systems we present the corresponding solutions of the functional (set-theoretic) Yang-Baxter equation, which are non-commutative versions of the maps arising from geometric crystals. Our approach works under additional condition of centrality of certain products of non-commuting variables. Then we apply such a restriction on the level of the Gel'fand-Dikii systems what allows to obtain non-autonomous (but with central non-autonomous factors) versions of the equations. In particular we recover known non-commutative version of Hirota's lattice sine-Gordon equation, and we present an integrable non-commutative and non-autonomous lattice modified Boussinesq equation.