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Published: 4 June 2021
Last updated: 20 February 2023

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1

CLARKSON, PETER A. "RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION." Analysis and Applications 06, no. 04 (October 2008): 349–69. http://dx.doi.org/10.1142/s0219530508001250.

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Rational solutions of the Boussinesq equation are expressed in terms of special polynomials associated with rational solutions of the second and fourth Painlevé equations, which arise as symmetry reductions of the Boussinesq equation. Further generalized rational solutions of the Boussinesq equation, which involve an infinite number of arbitrary constants, are derived. The generalized rational solutions are analogs of such solutions for the Korteweg–de Vries and nonlinear Schrödinger equations.
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2

Xu, Fei, Yixian Gao, and Weipeng Zhang. "Construction of Analytic Solution for Time-Fractional Boussinesq Equation Using Iterative Method." Advances in Mathematical Physics 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/506140.

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This paper is aimed at constructing analytical solution for both linear and nonlinear time-fractional Boussinesq equations by an iterative method. By the iterative process, we can obtain the analytic solution of the fourth-order time-fractional Boussinesq equation inR,R2, andRn, the sixth-order time-fractional Boussinesq equation, and the2nth-order time-fractional Boussinesq equation inR. Through these examples, it shows that the method is simple and effective.
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3

Clarkson, Peter A. "New exact solutions of the Boussinesq equation." European Journal of Applied Mathematics 1, no. 3 (September 1990): 279–300. http://dx.doi.org/10.1017/s095679250000022x.

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In this paper new exact solutions are derived for the physically and mathematically significant Boussinesq equation. These are obtained in two different ways: first, by generating exact solutions to the ordinary differential equations which arise from (classical and nonclassical) similarity reductions of the Boussinesq equation (these ordinary differential equations are solvable in terms of the first, second and fourth Painlevé equations); and second, by deriving new space-independent similarity reductions of the Boussinesq equation. Extensive sets of exact solutions for both the second and fourth Painlevé equations are also generated. The symbolic manipulation language MACSYMA is employed to facilitate the calculations involved.
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4

Jafari, Hossein, Nematollah Kadkhoda, and Chaudry Massod Khalique. "Application of Lie Symmetry Analysis and Simplest Equation Method for Finding Exact Solutions of Boussinesq Equations." Mathematical Problems in Engineering 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/452576.

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The Lie symmetry approach with simplest equation method is used to construct exact solutions of the bad Boussinesq and good Boussinesq equations. As the simplest equation, we have used the equation of Riccati.
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5

Fan, Fei, Bing Chen Liang, and Xiu Li Lv. "Study of Wave Models of Parabolic Mild Slope Equation and Boussinesq Equation." Applied Mechanics and Materials 204-208 (October 2012): 2334–40. http://dx.doi.org/10.4028/www.scientific.net/amm.204-208.2334.

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The Parabolic mild slope equation and Boussinesq equation wave models are studied in this paper. First, the wave models Funwave and REF/DIF, which based on Boussinesq equations and the parabolic mild slope equation, respectively, are introduced. And then, two experiments are used to study these two wave models, one is the non-breaking shoal experiment of University of Delaware and the other is the breaking undertow test experiment, which was finished in Ocean University of China by author. Last, the simulation data of two wave models are compared with the measured data. The results show that both Boussinesq equation and the parabolic mild slope equation wave models can simulated nearshore wave condition precisely, but Boussinesq equation wave models has a disadvantage in catching the variation of wave height caused by wave breaking.
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6

Bulut, Hasan, Münevver Tuz, and Tolga Akturk. "New Multiple Solution to the Boussinesq Equation and the Burgers-Like Equation." Journal of Applied Mathematics 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/952614.

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By considering an improved tanh function method, we found some exact solutions of Boussinesq and Burgers-like equations. The main idea of this method is to take full advantage of the Riccati equation which has more new solutions. We found some exact solutions of the Boussinesq equation and the Burgers-like equation.
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7

Rashidi, Saeede, and S. Reza Hejazi. "Symmetry properties, similarity reduction and exact solutions of fractional Boussinesq equation." International Journal of Geometric Methods in Modern Physics 14, no. 06 (May 4, 2017): 1750083. http://dx.doi.org/10.1142/s0219887817500839.

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In this paper, some properties of the time fractional Boussinesq equation are presented. Group analysis of the time fractional Boussinesq equation with Riemann–Liouville derivative is performed and the corresponding optimal system of subgroups are determined. Next, we apply the obtained optimal systems for constructing reduced fractional ordinary differential equations (FODEs). Finally, we show how to derive exact solutions to time fractional Boussinesq equation via invariant subspace method.
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8

Melinand, Benjamin. "Long wave approximation for water waves under a Coriolis forcing and the Ostrovsky equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 6 (July 19, 2018): 1201–37. http://dx.doi.org/10.1017/s0308210518000136.

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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.
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9

Johnson, R. S. "A Two-dimensional Boussinesq equation for water waves and some of its solutions." Journal of Fluid Mechanics 323 (September 25, 1996): 65–78. http://dx.doi.org/10.1017/s0022112096000845.

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A two-dimensional Boussinesq equation, \[u_{tt} - u_{xx} + 3(u^2)_{xx} - u_{xxxx} - u_{yy} = 0,\] is introduced to describe the propagation of gravity waves on the surface of water, in particular the head-on collision of oblique waves. This equation combines the two-way propagation of the classical Boussinesq equation with the (weak) dependence on a second spatial variable, as occurs in the two-dimensional Korteweg-de Vries (2D KdV) (or KPII) equation. Exact and general solitary-wave, two-soliton and resonant solutions are obtained from the Hirota bilinear form of the equation. The existence of a distributed-soliton solution is investigated, but it is shown that this is not a possibility. However the connection with the classical 2D KdV equation (which does possess such a solution) is explored via a suitable parametric representation of the dispersion relation.A three-soliton solution is also constructed, but this exists only if an auxiliary constraint among the six parameters is satisfied; thus the two-dimensional Boussinesq equation is not one of the class of completely integrable equations, confirming the analysis of Hietarinta (1987). This constraint is automatically satisfied for the classical Boussinesq equation (which is completely integrable). Graphical reproductions of some of the solutions of the two-dimensional Boussinesq equations are also presented.
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10

Abazari, Reza, and Adem Kılıçman. "Solitary Wave Solutions of the Boussinesq Equation and Its Improved Form." Mathematical Problems in Engineering 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/468206.

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This paper presents the general case study of previous works on generalized Boussinesq equations, (Abazari, 2011) and (Kılıcman and Abazari, 2012), that focuses on the application ofG′/G-expansion method with the aid of Maple to construct more general exact solutions for the coupled Boussinesq equations. In this work, the mentioned method is applied to construct more general exact solutions of Boussinesq equation and improved Boussinesq equation, which the French scientistJoseph Valentin Boussinesq(1842–1929) described in the 1870s model equations for the propagation of long waves on the surface of water with small amplitude. Our work is motivated by the fact that theG′/G-expansion method provides not only more general forms of solutions but also periodic, solitary waves and rational solutions. The method appears to be easier and faster by means of a symbolic computation.
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11

Alqufail, Mubarak A. H., M. S. A. Al-Amry, and Buthaina M. M. S. Own. "Analytic solutions for a new model of the(3+1)-Boussinesq equation." University of Aden Journal of Natural and Applied Sciences 23, no. 2 (October 31, 2019): 469–77. http://dx.doi.org/10.47372/uajnas.2019.n2.a17.

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In this paper, we have applied the mapping method to solve the (3+1)-dimensional Boussinenesq equation where we have obtained exact solutions for evolution equation to construct exact periodic and soliton solutions of nonlinear partial differential evolution equation. Many have obtained new families of exact traveling wave solutions, but the Boussinesq equation is successfully. These solutions may be significantly important for the explanation of some practical physical problems. New exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions and elliptic functions. It is shown that the mapping method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.
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12

Gandarias, M. L., and M. S. Bruzón. "Conservation laws for a Boussinesq equation." Applied Mathematics and Nonlinear Sciences 2, no. 2 (November 12, 2017): 465–72. http://dx.doi.org/10.21042/amns.2017.2.00037.

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AbstractIn this work, we study a generalized Boussinesq equation from the point of view of the Lie theory. We determine all the low-order conservation laws by using the multiplier method. Taking into account the relationship between symmetries and conservation laws and applying the multiplier method to a reduced ordinary differential equation, we obtain directly a second order ordinary differential equation and two third order ordinary differential equations.
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13

Bogdanov, L. V., and V. E. Zakharov. "The Boussinesq equation revisited." Physica D: Nonlinear Phenomena 165, no. 3-4 (May 2002): 137–62. http://dx.doi.org/10.1016/s0167-2789(02)00380-9.

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14

Wang, Deng-Shan, and Xiaodong Zhu. "Long-time asymptotics of the good Boussinesq equation with qxx-term and its modified version." Journal of Mathematical Physics 63, no. 12 (December 1, 2022): 123501. http://dx.doi.org/10.1063/5.0118374.

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Two modified Boussinesq equations along with their Lax pairs are proposed by introducing the Miura transformations. The modified good Boussinesq equation with initial condition is investigated by the Riemann–Hilbert method. Starting with the three-order Lax pair of this equation, the inverse scattering transform is formulated and the Riemann–Hilbert problem is established, and the properties of the reflection coefficients are presented. Then, the formulas of long-time asymptotics to the good Boussinesq equation and its modified version are given based on the Deift–Zhou approach of nonlinear steepest descent analysis. It is demonstrated that the results from the long-time asymptotic analysis are in excellent agreement with the numerical solutions. This is the first result on the long-time asymptotic behaviors of the good Boussinesq equation with q xx-term and its modified version.
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15

Young, William R. "Dynamic Enthalpy, Conservative Temperature, and the Seawater Boussinesq Approximation." Journal of Physical Oceanography 40, no. 2 (February 1, 2010): 394–400. http://dx.doi.org/10.1175/2009jpo4294.1.

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Abstract A new seawater Boussinesq system is introduced, and it is shown that this approximation to the equations of motion of a compressible binary solution has an energy conservation law that is a consistent approximation to the Bernoulli equation of the full system. The seawater Boussinesq approximation simplifies the mass conservation equation to ∇ · u = 0, employs the nonlinear equation of state of seawater to obtain the buoyancy force, and uses the conservative temperature introduced by McDougall as a thermal variable. The conserved energy consists of the kinetic energy plus the Boussinesq dynamic enthalpy h‡, which is the integral of the buoyancy with respect to geopotential height Z at a fixed conservative temperature and salinity. In the Boussinesq approximation, the full specific enthalpy h is the sum of four terms: McDougall’s potential enthalpy, minus the geopotential g0Z, plus the Boussinesq dynamic enthalpy h‡, and plus the dynamic pressure. The seawater Boussinesq approximation removes the large and dynamically inert contributions to h, and it reveals the important conversions between kinetic energy and h‡.
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16

Li, Biao, and Yong Chen. "Nonlinear Partial Differential Equations Solved by Projective Riccati Equations Ansatz." Zeitschrift für Naturforschung A 58, no. 9-10 (October 1, 2003): 511–19. http://dx.doi.org/10.1515/zna-2003-9-1007.

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Based on the general projective Riccati equations method and symbolic computation, some new exact travelling wave solutions are obtained for a nonlinear reaction-diffusion equation, the highorder modified Boussinesq equation and the variant Boussinesq equation. The obtained solutions contain solitary waves, singular solitary waves, periodic and rational solutions. From our results, we can not only recover the known solitary wave solutions of these equations found by existing various tanh methods and other sophisticated methods, but also obtain some new and more general travelling wave solutions.
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17

Ren, Bo. "Painlevé Analysis, Soliton Molecule, and Lump Solution of the Higher-Order Boussinesq Equation." Advances in Mathematical Physics 2021 (February 3, 2021): 1–6. http://dx.doi.org/10.1155/2021/6687632.

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The Painlevé integrability of the higher-order Boussinesq equation is proved by using the standard Weiss-Tabor-Carnevale (WTC) method. The multisoliton solutions of the higher-order Boussinesq equation are obtained by introducing dependent variable transformation. The soliton molecule and asymmetric soliton of the higher-order Boussinesq equation can be constructed by the velocity resonance mechanism. Lump solution can be derived by solving the bilinear form of the higher-order Boussinesq equation. By some detailed calculations, the lump wave of the higher-order Boussinesq equation is just the bright form. These types of the localized excitations are exhibited by selecting suitable parameters.
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18

Wayne, C. Eugene, and J. Douglas Wright. "Higher Order Modulation Equations for a Boussinesq Equation." SIAM Journal on Applied Dynamical Systems 1, no. 2 (January 2002): 271–302. http://dx.doi.org/10.1137/s1111111102411298.

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19

IVANOV, E., S. KRIVONOS, and R. P. MALIK. "N = 2 SUPER W3 ALGEBRA AND N = 2 SUPER BOUSSINESQ EQUATIONS." International Journal of Modern Physics A 10, no. 02 (January 20, 1995): 253–88. http://dx.doi.org/10.1142/s0217751x95000127.

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We study classical N=2 super W3 algebra and its interplay with N=2 supersymmetric extensions of the Boussinesq equation in the framework of the nonlinear realization method and the inverse Higgs-covariant reduction approach. These techniques have been previously used by us in the bosonic W3 case to give a new geometric interpretation of the Boussinesq hierarchy. Here we deduce the most general N=2 super Boussinesq equation and two kinds of the modified N=2 super Boussinesq equations, as well as the super Miura maps relating these systems to each other, by applying the covariant reduction to certain coset manifolds of linear [Formula: see text] symmetry associated with N=2 super W3. We discuss the integrability properties of the equations obtained and their correspondence with the formulation based on the notion of the second Hamiltonian structure.
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20

TÜLÜCE DEMİRAY, Şeyma, and Uğur BAYRAKCI. "Novel Solutions of Perturbed Boussinesq Equation." Journal of Mathematical Sciences and Modelling 5, no. 3 (December 1, 2022): 99–104. http://dx.doi.org/10.33187/jmsm.1123178.

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In this article, we have worked on the perturbed Boussinesq equation. We have applied the generalized Kudryashov method (GKM) and sine-Gordon expansion method (SGEM) to the perturbed Boussinesq equation. So, we have obtained some new soliton solutions of the perturbed Boussinesq equation. Furthermore, we have drawn some 2D and 3D graphics of these results by using Wolfram Mathematica 12.
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21

Hwang, Sooncheol, Sangyoung Son, and Patrick J. Lynett. "A GPU-ACCELERATED MODELING OF SCALAR TRANSPORT BASED ON BOUSSINESQ-TYPE EQUATIONS." Coastal Engineering Proceedings, no. 36v (December 28, 2020): 11. http://dx.doi.org/10.9753/icce.v36v.waves.11.

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This paper describes a two-dimensional scalar transport model solving advection-diffusion equation based on GPU-accelerated Boussinesq model called Celeris. Celeris is the firstly-developed Boussinesq-type model that is equipped with an interactive system between user and computing unit. Celeris provides greatly advantageous user-interface that one can change not only water level, topography but also model parameters while the simulation is running. In this study, an advection-diffusion equation for scalar transport was coupled with extended Boussinesq equations to simulate scalar transport in the nearshore.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/aHvMmdz3wps
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22

Wan, Renhui. "Long time stability for the dispersive SQG equation and Boussinesq equations in Sobolev space Hs." Communications in Contemporary Mathematics 22, no. 03 (September 10, 2018): 1850063. http://dx.doi.org/10.1142/s0219199718500633.

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Dispersive SQG equation have been studied by many works (see, e.g., [M. Cannone, C. Miao and L. Xue, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Londen. Math. Soc. 106 (2013) 650–674; T. M. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684; A. Kiselev and F. Nazarov, Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity 23 (2010) 549–554; R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys. 67 (2016) 104]), which is very similar to the 3D rotating Euler or Navier–Stokes equations. Long time stability for the dispersive SQG equation without dissipation was obtained by Elgindi–Widmayer [Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684], where the initial condition [Formula: see text] [Formula: see text] plays a important role in their proof. In this paper, by using the Strichartz estimate, we can remove this initial condition. Namely, we only assume the initial data is in the Sobolev space like [Formula: see text]. As an application, we can also obtain similar result for the 2D Boussinesq equations with the initial data near a nontrivial equilibrium.
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23

Moleleki, Letlhogonolo Daddy, and Chaudry Masood Khalique. "Solutions and Conservation Laws of a (2+1)-Dimensional Boussinesq Equation." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/548975.

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We study a nonlinear evolution partial differential equation, namely, the (2+1)-dimensional Boussinesq equation. For the first time Lie symmetry method together with simplest equation method is used to find the exact solutions of the (2+1)-dimensional Boussinesq equation. Furthermore, the new conservation theorem due to Ibragimov will be utilized to construct the conservation laws of the (2+1)-dimensional Boussinesq equation.
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24

Tao, Zhao-Ling. "Variational Principles for Some Nonlinear Wave Equations." Zeitschrift für Naturforschung A 63, no. 5-6 (June 1, 2008): 237–40. http://dx.doi.org/10.1515/zna-2008-5-601.

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Using the semi-inverse method proposed by Ji-Huan He, variational principles are established for some nonlinear wave equations arising in physics, including the Pochhammer-Chree equation, Zakharov-Kuznetsov equation, Korteweg-de Vries equation, Zhiber-Shabat equation, Kawahara equation, and Boussinesq equation.
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25

Jiang, Chaolong, Jianqiang Sun, Xunfeng He, and Lanlan Zhou. "High Order Energy-Preserving Method of the “Good” Boussinesq Equation." Numerical Mathematics: Theory, Methods and Applications 9, no. 1 (February 2016): 111–22. http://dx.doi.org/10.4208/nmtma.2015.m1420.

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AbstractThe fourth order average vector field (AVF) method is applied to solve the “Good” Boussinesq equation. The semi-discrete system of the “good” Boussinesq equation obtained by the pseudo-spectral method in spatial variable, which is a classical finite dimensional Hamiltonian system, is discretizated by the fourth order average vector field method. Thus, a new high order energy conservation scheme of the “good” Boussinesq equation is obtained. Numerical experiments confirm that the new high order scheme can preserve the discrete energy of the “good” Boussinesq equation exactly and simulate evolution of different solitary waves well.
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26

Lu, Jun-Feng. "Modified variational iteration method for variant Boussinesq equation." Thermal Science 19, no. 4 (2015): 1195–99. http://dx.doi.org/10.2298/tsci1504195l.

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In this paper, we solve the variant Boussinesq equation by the modified variational iteration method. The approximate solutions to the initial value problems of the variant Boussinesq equation are provided, and compared with the exact solutions. Numerical experiments show that the modified variational iteration method is efficient for solving the variant Boussinesq equation.
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27

Yuldashev, Tursun. "Mixed Boussinesq-Type Differential Equation." Vestnik Volgogradskogo gosudarstvennogo universiteta. Serija 1. Mathematica. Physica, no. 2 (June 2016): 13–26. http://dx.doi.org/10.15688/jvolsu1.2016.2.2.

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28

Airault, H. "Solutions of the Boussinesq equation." Physica D: Nonlinear Phenomena 21, no. 1 (August 1986): 171–76. http://dx.doi.org/10.1016/0167-2789(86)90088-6.

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29

Moleleki, Letlhogonolo Daddy, and Chaudry Masood Khalique. "Symmetries, Traveling Wave Solutions, and Conservation Laws of a(3+1)-Dimensional Boussinesq Equation." Advances in Mathematical Physics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/672679.

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We analyze the(3+1)-dimensional Boussinesq equation, which has applications in fluid mechanics. We find exact solutions of the(3+1)-dimensional Boussinesq equation by utilizing the Lie symmetry method along with the simplest equation method. The solutions obtained are traveling wave solutions. Moreover, we construct the conservation laws of the(3+1)-dimensional Boussinesq equation using the new conservation theorem, which is due to Ibragimov.
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30

Wazwaz, Abdul-Majid. "Gaussian solitary waves for the logarithmic Boussinesq equation and the logarithmic regularized Boussinesq equation." Ocean Engineering 94 (January 2015): 111–15. http://dx.doi.org/10.1016/j.oceaneng.2014.11.024.

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31

Hirota, Ryogo. "Solutions of the Classical Boussinesq Equation and the Spherical Boussinesq Equation: The Wronskian Technique." Journal of the Physical Society of Japan 55, no. 7 (July 15, 1986): 2137–50. http://dx.doi.org/10.1143/jpsj.55.2137.

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32

Tao, Sixing. "Nonlocal Symmetry, CRE Solvability, and Exact Interaction Solutions of the (2 + 1)-Dimensional Boussinesq Equation." Journal of Mathematics 2022 (April 23, 2022): 1–11. http://dx.doi.org/10.1155/2022/7850824.

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The (2 + 1)-dimensional Boussinesq equation is considered in this study. Nonlocal symmetries of the (2 + 1)-dimensional Boussinesq equation are obtained by means of the truncated Painlevé expansion. The consistent Riccati expansion (CRE) solvability of the Boussinesq equation is derived. Three special forms of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically.
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33

Yang, Zonghang, and Benny Y. C. Hon. "An Improved Modified Extended tanh-Function Method." Zeitschrift für Naturforschung A 61, no. 3-4 (April 1, 2006): 103–15. http://dx.doi.org/10.1515/zna-2006-3-401.

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In this paper we further improve the modified extended tanh-function method to obtain new exact solutions for nonlinear partial differential equations. Numerical applications of the proposed method are verified by solving the improved Boussinesq equation and the system of variant Boussinesq equations. The new exact solutions for these equations include Jacobi elliptic doubly periodic type,Weierstrass elliptic doubly periodic type, triangular type and solitary wave solutions
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34

Yang, Wei, and Chunguang Li. "General Propagation Lattice Boltzmann Model for the Boussinesq Equation." Entropy 24, no. 4 (March 30, 2022): 486. http://dx.doi.org/10.3390/e24040486.

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A general propagation lattice Boltzmann model is used to solve Boussinesq equations. Different local equilibrium distribution functions are selected, and the macroscopic equation is recovered with second order accuracy by means of the Chapman–Enskog multi-scale analysis and the Taylor expansion technique. To verify the effectiveness of the present model, some Boussinesq equations with initial boundary value problems are simulated. It is shown that our model can remain stable and accurate, which is an effective algorithm worthy of promotion and application.
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35

Aminikhah, Hossein. "Approximate analytical solution for the one-dimensional nonlinear Boussinesq equation." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 4 (May 5, 2015): 831–40. http://dx.doi.org/10.1108/hff-12-2013-0360.

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Purpose – The purpose of this paper is to provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. Combination of the Laplace transform and homotopy perturbation methods (LTHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation. Design/methodology/approach – The authors present the solution of nonlinear Boussinesq equation by combination of Laplace transform and new homotopy perturbation methods. An important property of the proposed method, which is clearly demonstrated in example, is that spectral accuracy is accessible in solving specific nonlinear nonlinear Boussinesq equation which has analytic solution functions. Findings – The authors proposed a combination of Laplace transform method and homotopy perturbation method to solve the one-dimensional Boussinesq equation. The results are found to be in excellent agreement. The results show that the LTNHPM is an effective mathematical tool which can play a very important role in nonlinear sciences. Originality/value – The authors provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. In this work combination of Laplace transform and new homotopy perturbation methods (LTNHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation.
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36

Xu, Xiao-Ge, Xiang-Hua Meng, and Qi-Xing Qu. "Lump soliton solutions and Bäcklund transformation for the (3+1)-dimensional Boussinesq equation with Bell polynomials." Modern Physics Letters B 32, no. 21 (July 26, 2018): 1850244. http://dx.doi.org/10.1142/s0217984918502445.

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In this paper, the (3+1)-dimensional Boussinesq equation which can describe the propagation of gravity waves on the surface of water is investigated. Using the Bell polynomials, the bilinear form of the (3+1)-dimensional Boussinesq equation is obtained and the lump soliton solutions for the equation are derived by means of the quadratic function method. As an important integrable property, the Bäcklund transformation for the (3+1)-dimensional Boussinesq equation is constructed by the Bell polynomials considering the constraints on the derivatives with respect to spatial and temporal variables. Through the relationship between the Bell polynomials and the Hirota bilinear operators, the bilinear Bäcklund transformation for the (3+1)-dimensional Boussinesq equation is given.
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37

Shakhmurov, Veli, and Rishad Shahmurov. "The regularity properties and blow-up of the solutions for improved Boussinesq equations." Electronic Journal of Qualitative Theory of Differential Equations, no. 89 (2021): 1–21. http://dx.doi.org/10.14232/ejqtde.2021.1.89.

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In this paper, we study the Cauchy problem for linear and nonlinear Boussinesq type equations that include the general differential operators. First, by virtue of the Fourier multipliers, embedding theorems in Sobolev and Besov spaces, the existence, uniqueness, and regularity properties of the solution of the Cauchy problem for the corresponding linear equation are established. Here, L p -estimates for a~solution with respect to space variables are obtained uniformly in time depending on the given data functions. Then, the estimates for the solution of linearized equation and perturbation of operators can be used to obtain the existence, uniqueness, regularity properties, and blow-up of solution at the finite time of the Cauchy for nonlinear for same classes of Boussinesq equations. Here, the existence, uniqueness, L p -regularity, and blow-up properties of the solution of the Cauchy problem for Boussinesq equations with differential operators coefficients are handled associated with the growth nature of symbols of these differential operators and their interrelationships. We can obtain the existence, uniqueness, and qualitative properties of different classes of improved Boussinesq equations by choosing the given differential operators, which occur in a wide variety of physical systems.
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38

Akinlar, Mehmet Ali, and Aydin Secer. "Wavelet-Petrov-Galerkin Method for Numerical Solution of Boussinesq Equation." Applied Mechanics and Materials 319 (May 2013): 451–55. http://dx.doi.org/10.4028/www.scientific.net/amm.319.451.

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In this paper, we employ the Wavelet-Petrov-Galerkin method to obtain the numerical solutions of the nonlinear Boussinesq equation. Boussinesq equation has braod application areas at different branches of engineering and science including chemistry and physics. We first discretize the Boussinesq equation in terms of wavelet coefficients and scaling functions, secondly multiply the discrete equation with wavelet basis functions. Using connection coefficients we express the resulting equation as a matrix equation. One of the significant advantages of the present method is that it does not require a quadrature formula.
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39

Park, H. M., and W. J. Lee. "Recursive Identification of Thermal Convection." Journal of Dynamic Systems, Measurement, and Control 125, no. 1 (March 1, 2003): 1–10. http://dx.doi.org/10.1115/1.1540116.

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A method is developed for the recursive identification of thermal convection system governed by the Boussinesq equation using an extended Kalman filter. A computationally feasible Kalman filter is constructed by reducing the Boussinesq equation to a small number of ordinary differential equations by means of the Karhunen-Loe`ve Galerkin procedure which is a type of Galerkin method employing the empirical eigenfunctions of the Karhunen-Loe`ve decomposition. Employing the Kalman filter constructed by using the reduced order model, the thermal convection induced by a spatially varying heat flux at the bottom is identified recursively by using either the Boussinesq equation or the reduced order model itself. The recursive identification technique developed in the present work is found to yield accurate results for thermal convection even with approximate covariance equation and noisy measurements. It is also shown that a reasonably accurate and computationally feasible method of recursive identification can be constructed even with a relatively inaccurate reduced order model.
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40

Wazwaz, Abdul-Majid. "A variety of soliton solutions for the Boussinesq-Burgers equation and the higher-order Boussinesq-Burgers equation." Filomat 31, no. 3 (2017): 831–40. http://dx.doi.org/10.2298/fil1703831w.

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In this work we examine the Boussinesq-Burgers equation and the higher-order Boussinesq- Burgers equation. The simplified Hirota?s method is used to derive multiple soliton solutions for each method. More soliton and periodic solutions are derived as well
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41

Bibi, Sadaf, Naveed Ahmed, Umar Khan, and Syed Tauseef Mohyud-Din. "Auxiliary equation method for ill-posed Boussinesq equation." Physica Scripta 94, no. 8 (May 24, 2019): 085213. http://dx.doi.org/10.1088/1402-4896/ab1951.

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42

Hikami, Kazuhiro. "The Baxter equation for quantum discrete Boussinesq equation." Nuclear Physics B 604, no. 3 (June 2001): 580–602. http://dx.doi.org/10.1016/s0550-3213(01)00204-8.

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43

ROY, P. K. "SUPERSYMMETRIC GENERALIZATION OF BOUSSINESQ–BURGER SYSTEM." International Journal of Modern Physics A 13, no. 10 (April 20, 1998): 1623–27. http://dx.doi.org/10.1142/s0217751x98000706.

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We generalize the Boussinesq–Burger coupled equation by constructing supersymmetric two boson system that is integrable. We obtain two super Hamiltonians and consequently two different set of supersymmetric coupled equations corresponding to the two Hamiltonians.
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44

Ren, Bo. "Characteristics of the Soliton Molecule and Lump Solution in the 2 + 1 -Dimensional Higher-Order Boussinesq Equation." Advances in Mathematical Physics 2021 (April 9, 2021): 1–7. http://dx.doi.org/10.1155/2021/5545984.

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The soliton molecules, as bound states of solitons, have attracted considerable attention in several areas. In this paper, the 2 + 1 -dimensional higher-order Boussinesq equation is constructed by introducing two high-order Hirota operators in the usual 2 + 1 -dimensional Boussinesq equation. By the velocity resonance mechanism, the soliton molecule and the asymmetric soliton of the higher-order Boussinesq equation are constructed. The soliton molecule does not exist for the usual 2 + 1 -dimensional Boussinesq equation. As a special kind of rational solution, the lump wave is localized in all directions and decays algebraically. The lump solution of the higher-order Boussinesq equation is obtained by using a quadratic function. This lump wave is just the bright form by some detail analysis. The graphics in this study are carried out by selecting appropriate parameters. The results in this work may enrich the variety of the dynamics of the high-dimensional nonlinear wave field.
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45

SHI Liang-ma, 史良马, 张世军 ZHANG Shi-jun, and 朱仁义 ZHU Ren-yi. "New Solutions to Boussinesq-Burgers Equation." Acta Sinica Quantum Optica 19, no. 1 (2013): 18–25. http://dx.doi.org/10.3788/asqo20131901.0018.

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46

Gandarias, María Luz, and María Rosa. "Symmetries and conservation laws of a damped Boussinesq equation." International Journal of Modern Physics B 30, no. 28n29 (November 10, 2016): 1640012. http://dx.doi.org/10.1142/s0217979216400129.

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In this work, we consider a damped equation with a time-independent source term. We derive the classical Lie symmetries admitted by the equation as well as the reduced ordinary differential equations. We also present some exact solutions. Conservation laws for this equation are constructed by using the multiplier method.
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47

Zeng, Libing, Keding Qin, and Shengqiang Tang. "Exact Explicit Traveling Wave Solution for the Generalized (2+1)-Dimensional Boussinesq Equation." ISRN Applied Mathematics 2011 (June 21, 2011): 1–10. http://dx.doi.org/10.5402/2011/419678.

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The sine-cosine method and the extended tanh method are used to construct exact solitary patterns solution and compactons solutions of the generalized (2+1)-dimensional Boussinesq equation. The compactons solutions and solitary patterns solutions of the generalized (2+1)-dimensional Boussinesq equation are successfully obtained. These solutions may be important and of significance for the explanation of some practical physical problems. It is shown that the sine-cosine and the extended tanh methods provide a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics.
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48

Demiray, Hilmi. "An Application of the Modified Reductive Perturbation Method to a Generalized Boussinesq Equation." International Journal of Nonlinear Sciences and Numerical Simulation 14, no. 1 (February 21, 2013): 27–31. http://dx.doi.org/10.1515/ijnsns-2011-0088.

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Abstract In this work, we apply “the modified reductive perturbation method” to the generalized Boussinesq equation and obtain various form of generalized KdV equations as the evolution equations. Seeking a localized travelling wave solutions for these evolution equations we determine the scale parameters g 1 and g 2, which corresponds to the correction terms in the wave speed, so as to remove the possible secularities that might occur. Depending on the sign and the values of certain parameters the resulting solutions are shown to be a solitary wave or a periodic solution. The suitability of the method is also shown by comparing the results with the exact travelling wave solution for the generalized Boussinesq equation.
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49

Song, Changming, Jina Li, and Ran Gao. "Nonexistence of Global Solutions to the Initial Boundary Value Problem for the Singularly Perturbed Sixth-Order Boussinesq-Type Equation." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/928148.

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We are concerned with the singularly perturbed Boussinesq-type equation including the singularly perturbed sixth-order Boussinesq equation, which describes the bidirectional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The nonexistence of global solution to the initial boundary value problem for the singularly perturbed Boussinesq-type equation is discussed and two examples are given.
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50

Mohamad- Jawad, Anwar. "The Sine-Cosine Function Method for Exact Solutions of Nonlinear Partial Differential Equations." Journal of Al-Rafidain University College For Sciences ( Print ISSN: 1681-6870 ,Online ISSN: 2790-2293 ), no. 2 (October 17, 2021): 120–39. http://dx.doi.org/10.55562/jrucs.v32i2.327.

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The Sine-Cosine function algorithm is applied for solving nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of nonlinear partial differential equations such as, The K(n + 1, n + 1) equation, Schrödinger-Hirota equation, Gardner equation, the modified KdV equation, perturbed Burgers equation, general Burger’s-Fisher equation, and Cubic modified Boussinesq equation which are the important Soliton equations.Keywords: Nonlinear PDEs, Exact Solutions, Nonlinear Waves, Gardner equation, Sine-Cosine function method, The Schrödinger-Hirota equation.
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