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Journal articles on the topic 'Kuznetsov equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 9 February 2022

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1

Li, Changzhao, and Juan Zhang. "Lie Symmetry Analysis and Exact Solutions of Generalized Fractional Zakharov-Kuznetsov Equations." Symmetry 11, no. 5 (2019): 601. http://dx.doi.org/10.3390/sym11050601.

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This paper considers the Lie symmetry analysis of a class of fractional Zakharov-Kuznetsov equations. We systematically show the procedure to obtain the Lie point symmetries for the equation. Accordingly, we study the vector fields of this equation. Meantime, the symmetry reductions of this equation are performed. Finally, by employing the obtained symmetry properties, we can get some new exact solutions to this fractional Zakharov-Kuznetsov equation.

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2

Kaltenbacher, Barbara, and Vanja Nikolić. "The Jordan–Moore–Gibson–Thompson Equation: Well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time." Mathematical Models and Methods in Applied Sciences 29, no. 13 (2019): 2523–56. http://dx.doi.org/10.1142/s0218202519500532.

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In this paper, we consider the Jordan–Moore–Gibson–Thompson equation, a third-order in time wave equation describing the nonlinear propagation of sound that avoids the infinite signal speed paradox of classical second-order in time strongly damped models of nonlinear acoustics, such as the Westervelt and the Kuznetsov equation. We show well-posedness in an acoustic velocity potential formulation with and without gradient nonlinearity, corresponding to the Kuznetsov and the Westervelt nonlinearities, respectively. Moreover, we consider the limit as the parameter of the third-order time derivati

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3

Khalique, Chaudry Masood. "Exact Explicit Solutions and Conservation Laws for a Coupled Zakharov-Kuznetsov System." Mathematical Problems in Engineering 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/461327.

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We study a coupled Zakharov-Kuznetsov system, which is an extension of a coupled Korteweg-de Vries system in the sense of the Zakharov-Kuznetsov equation. Firstly, we obtain some exact solutions of the coupled Zakharov-Kuznetsov system using the simplest equation method. Secondly, the conservation laws for the coupled Zakharov-Kuznetsov system will be constructed by using the multiplier approach.

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4

Rozanova, Anna. "The Khokhlov–Zabolotskaya–Kuznetsov equation." Comptes Rendus Mathematique 344, no. 5 (2007): 337–42. http://dx.doi.org/10.1016/j.crma.2007.01.010.

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5

Tao, Zhao-Ling. "Variational Principles for Some Nonlinear Wave Equations." Zeitschrift für Naturforschung A 63, no. 5-6 (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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6

Zayed, E. M. E., and K. A. E. Alurrfi. "The Generalized Projective Riccati Equations Method for Solving Nonlinear Evolution Equations in Mathematical Physics." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/259190.

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We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.

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7

Vinita and S. Saha Ray. "Symmetry analysis with similarity reduction, new exact solitary wave solutions and conservation laws of (3 + 1)-dimensional extended quantum Zakharov–Kuznetsov equation in quantum physics." Modern Physics Letters B 35, no. 09 (2021): 2150163. http://dx.doi.org/10.1142/s0217984921501633.

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A recently defined (3+1)-dimensional extended quantum Zakharov–Kuznetsov (QZK) equation is examined here by using the Lie symmetry approach. The Lie symmetry analysis has been used to obtain the varieties in invariant solutions of the extended Zakharov–Kuznetsov equation. Due to existence of arbitrary functions and constants, these solutions provide a rich physical structure. In this paper, the Lie point symmetries, geometric vector field, commutative table, symmetry groups of Lie algebra have been derived by using the Lie symmetry approach. The simplest equation method has been presented for

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8

Dekkers, Adrien, and Anna Rozanova-Pierrat. "Cauchy problem for the Kuznetsov equation." Discrete & Continuous Dynamical Systems - A 39, no. 1 (2019): 277–307. http://dx.doi.org/10.3934/dcds.2019012.

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9

Linares, Felipe, Mahendra Panthee, Tristan Robert, and Nikolay Tzvetkov. "On the periodic Zakharov-Kuznetsov equation." Discrete & Continuous Dynamical Systems - A 39, no. 6 (2019): 3521–33. http://dx.doi.org/10.3934/dcds.2019145.

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10

Cossetti, Lucrezia, Luca Fanelli, and Felipe Linares. "Uniqueness results for Zakharov-Kuznetsov equation." Communications in Partial Differential Equations 44, no. 6 (2019): 504–44. http://dx.doi.org/10.1080/03605302.2019.1581803.

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11

Wang, Guangming, and Zhong Han. "Some Reduction and Exact Solutions of a Higher-Dimensional Equation." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/597470.

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The conservation laws of the(3+1)-dimensional Zakharov-Kuznetsov equation were obtained using Noether’s theorem after an interesting substitutionu=vxto the equation. Then, with the aid of an obtained conservation law, the generalized double reduction theorem was applied to this equation. It can be verified that the reduced equation is a second order nonlinear ODE. Finally, some exact solutions of the Zakharov-Kuznetsov equation were constructed after solving the reduced equation.

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12

Ali, Muhammad Nasir, Aly R. Seadawy, and Syed Muhammad Husnine. "Lie point symmetries exact solutions and conservation laws of perturbed Zakharov–Kuznetsov equation with higher-order dispersion term." Modern Physics Letters A 34, no. 03 (2019): 1950027. http://dx.doi.org/10.1142/s0217732319500275.

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In this paper, Zakharov–Kuznetsov equation is investigated for exact solutions and conservation laws. The well-known Zakharov–Kuznetsov equation contains third-order dispersion, so its validity is restricted to the waves of small amplitudes only. When the amplitude of the wave increases, the velocity and the width of the soliton deviate from the prediction of this equation. To overcome this deficiency, higher-order dispersion term is added to the Zakharov–Kuznetsov equation. We obtain Lie point symmetries and conservation laws for this new model. Wave transformation is applied to convert the n

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13

Yin, Xiao-Jun, Quan-Sheng Liu, Lian-Gui Yang, and N. Narenmandula. "The new exact solitary solutions for the (3+1)-dimensional Zakharov-Kuznetsov equation using the Riccati equation." Thermal Science 24, no. 6 Part B (2020): 3995–4000. http://dx.doi.org/10.2298/tsci2006995y.

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In this paper, a non-linear (3+1)-dimensional Zakharov-Kuznetsov equation is investigated by employing the subsidiary equation method, which arises in quantum magneto plasma. The periodic solutions, rational wave solutions, soliton solutions for the quantum Zakharov-Kuznetsov equation which play an important role in mathematical physics are obtained with the help of the Riccati equation expan?sion method. Meanwhile, the electrostatic potential can be accordingly obtained. Compared to the other methods, the exact solutions obtained will extend on earlier reports by using the Riccati equation.

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14

Ray, S. Saha, and S. Sahoo. "New Exact Solutions of Fractional Zakharov—Kuznetsov and Modified Zakharov—Kuznetsov Equations Using Fractional Sub-Equation Method." Communications in Theoretical Physics 63, no. 1 (2015): 25–30. http://dx.doi.org/10.1088/0253-6102/63/1/05.

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15

Jiang, Jun, Yuqiang Feng, and Shougui Li. "Improved Fractional Subequation Method and Exact Solutions to Fractional Partial Differential Equations." Journal of Function Spaces 2020 (May 17, 2020): 1–18. http://dx.doi.org/10.1155/2020/5840920.

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In this paper, the improved fractional subequation method is applied to establish the exact solutions for some nonlinear fractional partial differential equations. Solutions to the generalized time fractional biological population model, the generalized time fractional compound KdV-Burgers equation, the space-time fractional regularized long-wave equation, and the (3+1)-space-time fractional Zakharov-Kuznetsov equation are obtained, respectively.

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16

Zayed, EL Sayed M. E., and Abdul-Ghani Al-Nowehy. "Exact solutions of the Biswas-Milovic equation, the ZK(m,n,k) equation and the K(m,n) equation using the generalized Kudryashov method." Open Physics 14, no. 1 (2016): 129–39. http://dx.doi.org/10.1515/phys-2016-0013.

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AbstractIn this article, we apply the generalized Kudryashov method for finding exact solutions of three nonlinear partial differential equations (PDEs), namely: the Biswas-Milovic equation with dual-power law nonlinearity; the Zakharov--Kuznetsov equation (ZK(m,n,k)); and the K(m,n) equation with the generalized evolution term. As a result, many analytical exact solutions are obtained including symmetrical Fibonacci function solutions, and hyperbolic function solutions. Physical explanations for certain solutions of the three nonlinear PDEs are obtained.

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17

Yang, Shu. "Exact Solutions to Zakharov-Kuznetsov Equation with Variable Coefficients by Trial Equation Method." Zeitschrift für Naturforschung A 73, no. 1 (2017): 1–4. http://dx.doi.org/10.1515/zna-2017-0269.

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AbstractBy the trial equation method and the complete discrimination system for polynomial method, some exact solutions to Zakharov-Kuznetsov equation with variable coefficients are obtained. These solutions include solitary solutions, rational solutions, periodic solution and double periodic solutions.

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18

Rezazadeh, Hadi, Javad Vahidi, Asim Zafar, and Ahmet Bekir. "The Functional Variable Method to Find New Exact Solutions of the Nonlinear Evolution Equations with Dual-Power-Law Nonlinearity." International Journal of Nonlinear Sciences and Numerical Simulation 21, no. 3-4 (2020): 249–57. http://dx.doi.org/10.1515/ijnsns-2019-0064.

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AbstractIn this work, we established new travelling wave solutions for some nonlinear evolution equations with dual-power-law nonlinearity namely the Zakharov–Kuznetsov equation, the Benjamin–Bona–Mahony equation and the Korteweg–de Vries equation. The functional variable method was used to construct travelling wave solutions of nonlinear evolution equations with dual-power-law nonlinearity. The travelling wave solutions are expressed by generalized hyperbolic functions and the rational functions. This method presents a wider applicability for handling nonlinear wave equations.

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19

Shivamoggi, B. K., D. K. Rollins, and R. Fanjul. "Analytic aspects of the Zakharov–Kuznetsov equation." Physica Scripta 47, no. 1 (1993): 15–17. http://dx.doi.org/10.1088/0031-8949/47/1/003.

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20

Heng-Chun, Hu. "New Exact Solutions of Zakharov–Kuznetsov Equation." Communications in Theoretical Physics 49, no. 3 (2008): 559–61. http://dx.doi.org/10.1088/0253-6102/49/3/07.

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21

Changzheng, Qu. "Nonclassical analysis for the Zakharov-Kuznetsov equation." International Journal of Theoretical Physics 34, no. 1 (1995): 99–108. http://dx.doi.org/10.1007/bf00670990.

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22

DEHGHAN, MEHDI, JALIL MANAFIAN, and ABBAS SAADATMANDI. "ANALYTICAL TREATMENT OF SOME PARTIAL DIFFERENTIAL EQUATIONS ARISING IN MATHEMATICAL PHYSICS BY USING THEExp-FUNCTION METHOD." International Journal of Modern Physics B 25, no. 22 (2011): 2965–81. http://dx.doi.org/10.1142/s021797921110148x.

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The Exp -function method with the aid of symbolic computational system can be used to obtain the generalized solitary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics. In this paper, we study the analytic treatment of the Zakharov–Kuznetsov (ZK) equation, the modified ZK equation, and the generalized forms of these equations. Exact solutions with solitons and periodic structures are obtained.

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23

Javeed, Shumaila, Khurram Saleem Alimgeer, Sidra Nawaz, et al. "Soliton Solutions of Mathematical Physics Models Using the Exponential Function Technique." Symmetry 12, no. 1 (2020): 176. http://dx.doi.org/10.3390/sym12010176.

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This paper is based on finding the exact solutions for Burger’s equation, Zakharov-Kuznetsov (ZK) equation and Kortewegde vries (KdV) equation by utilizing exponential function method that depends on the series of exponential functions. The exponential function method utilizes the homogeneous balancing principle to find the solutions of nonlinear equations. This method is simple, wide-reaching and helpful for finding the exact solution of nonlinear conformable PDEs.

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24

Naher, Hasibun, and Farah Aini Abdullah. "New Traveling Wave Solutions by the Extended Generalized Riccati Equation Mapping Method of the(2+1)-Dimensional Evolution Equation." Journal of Applied Mathematics 2012 (2012): 1–18. http://dx.doi.org/10.1155/2012/486458.

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The generalized Riccati equation mapping is extended with the basic(G′/G)-expansion method which is powerful and straightforward mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty-seven traveling wave solutions for the (2+1)-dimensional modified Zakharov-Kuznetsov equation by applying this method. Further, the auxiliary equationG′(η)=w+uG(η)+vG2(η)is executed with arbitrary constant coefficients and called the generalized Riccati equation. The obtained solutions including solitons and periodic solutions are illustrated through the hyperbo

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25

Khan, Kamruzzaman, M. Ali Akbar, and Norhashidah Hj Mohd Ali. "The Modified Simple Equation Method for Exact and Solitary Wave Solutions of Nonlinear Evolution Equation: The GZK-BBM Equation and Right-Handed Noncommutative Burgers Equations." ISRN Mathematical Physics 2013 (February 25, 2013): 1–5. http://dx.doi.org/10.1155/2013/146704.

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The modified simple equation method is significant for finding the exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. In this paper, we bring in the modified simple equation (MSE) method for solving NLEEs via the Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony (GZK-BBM) equation and the right-handed noncommutative Burgers' (nc-Burgers) equations and achieve the exact solutions involving parameters. When the parameters are taken as special values, the solitary wave solutions are originated from the traveling wave solutions. It is established tha

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26

Yu, Jianping, Deng-Shan Wang, Yongli Sun, and Suping Wu. "Modified method of simplest equation for obtaining exact solutions of the Zakharov–Kuznetsov equation, the modified Zakharov–Kuznetsov equation, and their generalized forms." Nonlinear Dynamics 85, no. 4 (2016): 2449–65. http://dx.doi.org/10.1007/s11071-016-2837-7.

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27

Akbar, M. Ali, Norhashidah Hj Mohd Ali, and E. M. E. Zayed. "A Generalized and Improved(G′/G)-Expansion Method for Nonlinear Evolution Equations." Mathematical Problems in Engineering 2012 (2012): 1–22. http://dx.doi.org/10.1155/2012/459879.

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A generalized and improved(G′/G)-expansion method is proposed for finding more general type and new travelling wave solutions of nonlinear evolution equations. To illustrate the novelty and advantage of the proposed method, we solve the KdV equation, the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation and the strain wave equation in microstructured solids. Abundant exact travelling wave solutions of these equations are obtained, which include the soliton, the hyperbolic function, the trigonometric function, and the rational functions. Also it is shown that the proposed method is effic

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28

Ma, Hong-Cai, Yao-Dong Yu, and Dong-Jie Ge. "The auxiliary equation method for solving the Zakharov–Kuznetsov (ZK) equation." Computers & Mathematics with Applications 58, no. 11-12 (2009): 2523–27. http://dx.doi.org/10.1016/j.camwa.2009.03.036.

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29

Nawaz, Rashid, Laiq Zada, Abraiz Khattak, Muhammad Jibran, and Adam Khan. "Optimum Solutions of Fractional Order Zakharov–Kuznetsov Equations." Complexity 2019 (December 10, 2019): 1–9. http://dx.doi.org/10.1155/2019/1741958.

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In this paper, the Optimal Homotopy Asymptotic Method is extended to derive the approximate solutions of fractional order two-dimensional partial differential equations. The fractional order Zakharov–Kuznetsov equation is solved as a test example, while the time fractional derivatives are described in the Caputo sense. The solutions of the problem are computed in the form of rapidly convergent series with easily calculable components using Mathematica. Reliability of the proposed method is given by comparison with other methods in the literature. The obtained results showed that the method is

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30

Li, Haochen, Jianqiang Sun, and Mengzhao Qin. "Multi-Symplectic Method for the Zakharov-Kuznetsov Equation." Advances in Applied Mathematics and Mechanics 7, no. 1 (2015): 58–73. http://dx.doi.org/10.4208/aamm.2013.m128.

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AbstractA new scheme for the Zakharov-Kuznetsov (ZK) equation with the accuracy order of is proposed. The multi-symplectic conservation property of the new scheme is proved. The backward error analysis of the new multi-symplectic scheme is also implemented. The solitary wave evolution behaviors of the Zakharov-Kunetsov equation is investigated by the new multi-symplectic scheme. The accuracy of the scheme is analyzed.

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31

Chen, Mo, and Lionel Rosier. "Exact controllability of the linear Zakharov-Kuznetsov equation." Discrete & Continuous Dynamical Systems - B 25, no. 10 (2020): 3889–916. http://dx.doi.org/10.3934/dcdsb.2020080.

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32

Shivamoggi, Bhimsen K. "The Painlevé analysis of the Zakharov-Kuznetsov equation." Physica Scripta 42, no. 6 (1990): 641–42. http://dx.doi.org/10.1088/0031-8949/42/6/001.

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33

Cleveland, Robin O. "Shocking stones with the Khokhlov–Zabolotskaya–Kuznetsov equation." Journal of the Acoustical Society of America 126, no. 4 (2009): 2202. http://dx.doi.org/10.1121/1.3248620.

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34

Chen, Mo. "Unique continuation property for the Zakharov–Kuznetsov equation." Computers & Mathematics with Applications 77, no. 5 (2019): 1273–81. http://dx.doi.org/10.1016/j.camwa.2018.11.002.

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35

Bustamante, Eddye, José Jiménez Urrea, and Jorge Mejía. "The Zakharov–Kuznetsov equation in weighted Sobolev spaces." Journal of Mathematical Analysis and Applications 433, no. 1 (2016): 149–75. http://dx.doi.org/10.1016/j.jmaa.2015.07.024.

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36

Iwasaki, Hiroshi, Sadayoshi Toh, and Takuji Kawahara. "Cylindrical quasi-solitons of the Zakharov-Kuznetsov equation." Physica D: Nonlinear Phenomena 43, no. 2-3 (1990): 293–303. http://dx.doi.org/10.1016/0167-2789(90)90138-f.

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37

Zhao, Xiaoshan, Hongxian Zhou, Yaning Tang, and Huabing Jia. "Travelling wave solutions for modified Zakharov–Kuznetsov equation." Applied Mathematics and Computation 181, no. 1 (2006): 634–48. http://dx.doi.org/10.1016/j.amc.2006.01.049.

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38

Nadjafikhah, Mehdi, and Fatemeh Ahangari. "SYMMETRY ANALYSIS AND SIMILARITY REDUCTION OF THE KORTEWEG–DE VRIES–ZAKHAROV–KUZNETSOV EQUATION." Asian-European Journal of Mathematics 05, no. 01 (2012): 1250006. http://dx.doi.org/10.1142/s1793557112500064.

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In this paper, the problem of determining the largest possible set of symmetries for an important nonlinear dynamical system in mathematical physics, the Korteweg–de Vries–Zakharov–Kuznetsov (KdV–ZK) equation, is studied. By applying the basic Lie symmetry method for the KdV–ZK equation, the classical Lie point symmetry operators are obtained. Also, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras of the equation is constructed. The Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained. The

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39

Islam, M. Nurul, Rehana Parvin, Mst Rashida Pervin, and M. Ali Akbar. "Adequate soliton solutions to the time fractional Zakharov-Kuznetsov equation and the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation." Arab Journal of Basic and Applied Sciences 28, no. 1 (2021): 370–85. http://dx.doi.org/10.1080/25765299.2021.1969740.

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40

Li, Changzhao, and Hui Fang. "Stochastic Bifurcations of Group-Invariant Solutions for a Generalized Stochastic Zakharov–Kuznetsov Equation." International Journal of Bifurcation and Chaos 31, no. 03 (2021): 2150040. http://dx.doi.org/10.1142/s0218127421500401.

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In this paper, we introduce the concept of stochastic bifurcations of group-invariant solutions for stochastic nonlinear wave equations. The essence of this concept is to display bifurcation phenomena by investigating stochastic P-bifurcation and stochastic D-bifurcation of stochastic ordinary differential equations derived by Lie symmetry reductions of stochastic nonlinear wave equations. Stochastic bifurcations of group-invariant solutions can be considered as an indirect display of bifurcation phenomena of stochastic nonlinear wave equations. As a constructive example, we study stochastic b

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41

Zhang, Wenbin, and Jiangbo Zhou. "Traveling Wave Solutions of a Generalized Zakharov-Kuznetsov Equation." ISRN Mathematical Analysis 2012 (March 13, 2012): 1–10. http://dx.doi.org/10.5402/2012/107846.

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We employ the bifurcation theory of planar dynamical system to investigate the traveling-wave solutions of the generalized Zakharov-Kuznetsov equation. Four important types of traveling wave solutions are obtained, which include the solitary wave solutions, periodic solutions, kink solutions, and antikink solutions.

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42

Abdul-Wahab, Mohammed Sabah, and A. S. J. Al-Saif. "A New Technique for Simulation the Zakharov–Kuznetsov Equation." JOURNAL OF ADVANCES IN MATHEMATICS 14, no. 2 (2018): 7912–20. http://dx.doi.org/10.24297/jam.v14i2.7559.

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In this article, a new technique is proposed to simulated two-dimensional Zakharov–Kuznetsov equation with the initial condition. The idea of this technique is based on Taylors' series in its derivation. Two test problems are presented to illustrate the performance of the new scheme. Analytical approximate solutions that we obtain are compared with variational iteration method (VIM) and homotopy analysis method (HAM). The results show that the new scheme is efficient and better than the other methods in accuracy and convergence.

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43

Shan, Minjie. "Well-Posedness for the Two-Dimensional Zakharov-Kuznetsov Equation." Funkcialaj Ekvacioj 63, no. 1 (2020): 67–95. http://dx.doi.org/10.1619/fesi.63.67.

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44

Linares, Felipe, and Jean-Claude Saut. "The Cauchy problem for the 3D Zakharov-Kuznetsov equation." Discrete & Continuous Dynamical Systems - A 24, no. 2 (2009): 547–65. http://dx.doi.org/10.3934/dcds.2009.24.547.

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45

Qian, Xu, Song-He Song, Er Gao, and Wei-Bin Li. "Explicit multi-symplectic method for the Zakharov—Kuznetsov equation." Chinese Physics B 21, no. 7 (2012): 070206. http://dx.doi.org/10.1088/1674-1056/21/7/070206.

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46

Saravi, Masoud, and Ali Nikkar. "An efficient iterative method for solving Zakharov-Kuznetsov Equation." Journal of Physics: Conference Series 474 (November 29, 2013): 012029. http://dx.doi.org/10.1088/1742-6596/474/1/012029.

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47

Murawski, K., and P. M. Edwin. "The Zakharov–Kuznetsov equation for nonlinear ion-acoustic waves." Journal of Plasma Physics 47, no. 1 (1992): 75–83. http://dx.doi.org/10.1017/s0022377800024090.

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The Zakharov-Kuznetsov equation is used to describe ion-acoustic wave propagation in a magnetic environment. An initial-value problem is solved for this equation on the basis of a numerical method that uses the fast-Fourier-transform technique for calculating space derivatives and a fourth-order Runge-Kutta method for the time scheme. Numerical simulations show that the disturbed flat (planar) solitary waves can break up into more robust cylindrical ones. Interactions between these two types of wave, and recurrence phenomena, are also studied.

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48

ROZANOVA-PIERRAT, ANNA. "QUALITATIVE ANALYSIS OF THE KHOKHLOV–ZABOLOTSKAYA–KUZNETSOV (KZK) EQUATION." Mathematical Models and Methods in Applied Sciences 18, no. 05 (2008): 781–812. http://dx.doi.org/10.1142/s0218202508002863.

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The Khokhlov–Zabolotskaya–Kuznetzov (KZK) equation is considered as a model of nonlinear acoustic which describes the nonlinear propagation of a finite-amplitude focused sound beam which is essentially one-directional, in the thermo-viscous medium.1,7,8 The aim of this paper is the study of the existence, uniqueness, stability, regularity, continuous dependence on the initial value and blow-up of solution of the KZK equation in Sobolev spaces Hs of periodic on x functions and with mean value zero. Existence of shock waves for the model with zero viscosity is proved using S. Alinhac's method.2

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Larkine, Nikolai Andreevitch. "The 2D Zakharov-Kuznetsov-Burgers equation on a strip." Boletim da Sociedade Paranaense de Matemática 34, no. 1 (2016): 151–72. http://dx.doi.org/10.5269/bspm.v34i1.26049.

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Abstract:

An initial-boundary value problem for the 2D Zakharov-Kuznetsov-Burgers equation posed on a channel-type strip was considered. The existence and uniqueness results for regular and weak solutions in weighted spaces as well as exponential decay of small solutions without restrictions on the width of a strip were proven both for regular solutions in an elevated norm and for weak solutions in the $L^2$-norm.

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Peng, Yan-Ze. "Exact travelling wave solutions for the Zakharov–Kuznetsov equation." Applied Mathematics and Computation 199, no. 2 (2008): 397–405. http://dx.doi.org/10.1016/j.amc.2007.08.095.

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