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Journal articles on the topic 'GKdV'

Author: Grafiati

Published: 4 June 2021

Last updated: 3 February 2022

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1

Martel, Yvan, and Frank Merle. "Review of long time asymptotics and collision of solitons for the quartic generalized Korteweg—de Vries equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 141, no. 2 (April 2011): 287–317. http://dx.doi.org/10.1017/s030821051000003x.

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We review recent nonlinear partial differential equation techniques developed to address questions concerning solitons for the quartic generalized Korteweg—de Vries equation (gKdV) and other generalizations of the KdV equation. We draw a comparison between results obtained in this way and some elements of the classical integrability theory for the original KdV equation, which serve as a reference for soliton and multi-soliton problems. First, known results on stability and asymptotic stability of solitons for gKdV equations are reviewed from several different sources. Second, we consider the problem of the interaction of two solitons for the quartic gKdV equation. We focus on recent results and techniques from a previous paper by the present authors concerning the interaction of two almost-equal solitons.

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2

Himonas, A. Alexandrou, and Gerson Petronilho. "Analytic well-posedness of periodic gKdV." Journal of Differential Equations 253, no. 11 (December 2012): 3101–12. http://dx.doi.org/10.1016/j.jde.2012.08.024.

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3

ZHENG, CHUN-LONG, HAI-PING ZHU, and JIAN-PING FANG. "FRACTAL AND CHAOTIC PATTERNS OF GENERAL KORTEWEG–DE VRIES SYSTEM IN (2+1)-DIMENSIONS." International Journal of Modern Physics B 20, no. 28 (November 10, 2006): 4843–54. http://dx.doi.org/10.1142/s0217979206035539.

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With the aid of an extended projective method and a variable separation approach, new families of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for (2+1)-dimensional general Korteweg–de Vries (GKdV) system are derived. Analytical investigation of the (2+1)-dimensional GKdV system shows the existence of abundant stable localized coherent excitations such as dromions, lumps, peakons, compactons and ring soliton solutions as well as rich fractal and chaotic localized patterns in terms of the derived solitary solutions or the variable separation solutions when we consider appropriate boundary conditions and/or initial qualifications.

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4

Bagheri, Majid, and Ali Khani. "Analytical Method for Solving the Fractional Order Generalized KdV Equation by a Beta-Fractional Derivative." Advances in Mathematical Physics 2020 (November 4, 2020): 1–18. http://dx.doi.org/10.1155/2020/8819183.

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The present work is related to solving the fractional generalized Korteweg-de Vries (gKdV) equation in fractional time derivative form of order α . Some exact solutions of the fractional-order gKdV equation are attained by employing the new powerful expansion approach by using the beta-fractional derivative which is used to get many solitary wave solutions by changing various parameters. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of nonlinearity. Some of the nonlinear equations arise in fluid dynamics and nonlinear phenomena.

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5

Strunk, Nils. "Well-posedness for the supercritical gKdV equation." Communications on Pure and Applied Analysis 13, no. 2 (October 2013): 527–42. http://dx.doi.org/10.3934/cpaa.2014.13.527.

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6

Martel, Yvan, and Frank Merle. "Refined asymptotics around solitons for gKdV equations." Discrete & Continuous Dynamical Systems - A 20, no. 2 (2008): 177–218. http://dx.doi.org/10.3934/dcds.2008.20.177.

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7

Koch, Herbert. "Self-similar solutions to super-critical gKdV." Nonlinearity 28, no. 3 (January 27, 2015): 545–75. http://dx.doi.org/10.1088/0951-7715/28/3/545.

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8

Hannah, Heather, A. Alexandrou Himonas, and Gerson Petronilho. "Gevrey regularity of the periodic gKdV equation." Journal of Differential Equations 250, no. 5 (March 2011): 2581–600. http://dx.doi.org/10.1016/j.jde.2010.12.020.

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9

Wang, Ming, Dongfang Li, Chengjian Zhang, and Yanbin Tang. "Long time behavior of solutions of gKdV equations." Journal of Mathematical Analysis and Applications 390, no. 1 (June 2012): 136–50. http://dx.doi.org/10.1016/j.jmaa.2012.01.031.

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10

Combet, Vianney. "Multi-Soliton Solutions for the Supercritical gKdV Equations." Communications in Partial Differential Equations 36, no. 3 (December 28, 2010): 380–419. http://dx.doi.org/10.1080/03605302.2010.503770.

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11

Martel, Yvan, Frank Merle, and Tai-Peng Tsai. "Stability and Asymptotic Stability for Subcritical gKdV Equations." Communications in Mathematical Physics 231, no. 2 (December 1, 2002): 347–73. http://dx.doi.org/10.1007/s00220-002-0723-2.

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12

Omel’yanov, Georgy. "Wave Collision for the gKdV-4 equation. Asymptotic Approach." Interdisciplinary journal of Discontinuity, Nonlinearity and Complexity 6, no. 1 (March 2017): 35–47. http://dx.doi.org/10.5890/dnc.2017.03.004.

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13

Combet, Vianney, and Yvan Martel. "Construction of Multibubble Solutions for the Critical GKDV Equation." SIAM Journal on Mathematical Analysis 50, no. 4 (January 2018): 3715–90. http://dx.doi.org/10.1137/17m1140595.

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14

Muñoz, Claudio. "Inelastic Character of Solitons of Slowly Varying gKdV Equations." Communications in Mathematical Physics 314, no. 3 (March 31, 2012): 817–52. http://dx.doi.org/10.1007/s00220-012-1463-6.

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15

Martel, Yvan, Frank Merle, Kenji Nakanishi, and Pierre Raphaël. "Codimension One Threshold Manifold for the Critical gKdV Equation." Communications in Mathematical Physics 342, no. 3 (November 23, 2015): 1075–106. http://dx.doi.org/10.1007/s00220-015-2509-3.

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16

Martel, Yvan, and Frank Merle. "Stability of Two Soliton Collision for Nonintegrable gKdV Equations." Communications in Mathematical Physics 286, no. 1 (November 21, 2008): 39–79. http://dx.doi.org/10.1007/s00220-008-0685-0.

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17

Martel, Yvan, and Frank Merle. "Description of two soliton collision for the quartic gKdV equation." Annals of Mathematics 174, no. 2 (September 1, 2011): 757–857. http://dx.doi.org/10.4007/annals.2011.174.2.2.

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18

Lan, Yang. "Blow-up solutions forL2supercritical gKdV equations with exactlykblow-up points." Nonlinearity 30, no. 8 (July 12, 2017): 3203–40. http://dx.doi.org/10.1088/1361-6544/aa7765.

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19

Martel, Yvan, and Frank Merle. "Asymptotic stability of solitons of the subcritical gKdV equations revisited." Nonlinearity 18, no. 1 (October 2, 2004): 55–80. http://dx.doi.org/10.1088/0951-7715/18/1/004.

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20

Lai, X., and Q. Cao. "Some finite difference methods for a kind of GKdV equations." Communications in Numerical Methods in Engineering 23, no. 3 (June 29, 2006): 179–96. http://dx.doi.org/10.1002/cnm.889.

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21

Jin, Jiayin, Zhiwu Lin, and Chongchun Zeng. "Dynamics near the solitary waves of the supercritical gKDV equations." Journal of Differential Equations 267, no. 12 (December 2019): 7213–62. http://dx.doi.org/10.1016/j.jde.2019.07.019.

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22

Farah, Luiz G., and Ademir Pastor. "On well-posedness and wave operator for the gKdV equation." Bulletin des Sciences Mathématiques 137, no. 3 (April 2013): 229–41. http://dx.doi.org/10.1016/j.bulsci.2012.04.002.

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23

Richards, Geordie. "Invariance of the Gibbs measure for the periodic quartic gKdV." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 33, no. 3 (May 2016): 699–766. http://dx.doi.org/10.1016/j.anihpc.2015.01.003.

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24

Rab, M. Abdur, and Jasmin Akhter. "Sine-Function Method In The Soliton Solution Of Nonlinear Partial Differential Equations." GANIT: Journal of Bangladesh Mathematical Society 32 (February 4, 2013): 55–60. http://dx.doi.org/10.3329/ganit.v32i0.13647.

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In this paper we establish a traveling wave solution for nonlinear partial differential equations using sine-function method. The method is used to obtain the exact solutions for three different types of nonlinear partial differential equations like general equal width wave equation (GEWE), general regularized long wave equation (GRLW), general Korteweg-de Vries equation(GKDV) which are the important soliton equations DOI: http://dx.doi.org/10.3329/ganit.v32i0.13647 GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 32 (2012) 55-60

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25

Kocak, Huseyin. "Kink and anti-kink wave solutions for the generalized KdV equation with Fisher-type nonlinearity." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 11, no. 2 (April 2, 2021): 123–27. http://dx.doi.org/10.11121/ijocta.01.2021.00973.

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This paper proposes a new dispersion-convection-reaction model, which is called the gKdV-Fisher equation, to obtain the travelling wave solutions by using the Riccati equation method. The proposed equation is a third-order dispersive partial differential equation combining the purely nonlinear convective term with the purely nonlinear reactive term. The obtained global and blow-up solutions, which might be used in the further numerical and analytical analyses of such models, are illustrated with suitable parameters.

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26

Martel, Yvan, Frank Merle, and Pierre Raphaël. "Blow up for the critical gKdV equation. II: Minimal mass dynamics." Journal of the European Mathematical Society 17, no. 8 (2015): 1855–925. http://dx.doi.org/10.4171/jems/547.

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27

Wang, Ming. "Global attractor for weakly damped gKdV equations in higher sobolev spaces." Discrete & Continuous Dynamical Systems - A 35, no. 8 (2015): 3799–825. http://dx.doi.org/10.3934/dcds.2015.35.3799.

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28

Huang, Lie-de. "THE NECESSARY CONDITION FOR SHORT RANGE SOLITON SOLUTION OF GKdV EQUATION." Acta Mathematica Scientia 11, no. 2 (April 1991): 209–12. http://dx.doi.org/10.1016/s0252-9602(18)30233-9.

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29

Wu, Ranchao, and Jianhua Sun. "Soliton-like solutions to the GKdV equation by extended mapping method." Chaos, Solitons & Fractals 31, no. 1 (January 2007): 70–74. http://dx.doi.org/10.1016/j.chaos.2005.09.032.

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30

Martel, Yvan, and Frank Merle. "Inelastic interaction of nearly equal solitons for the quartic gKdV equation." Inventiones mathematicae 183, no. 3 (September 17, 2010): 563–648. http://dx.doi.org/10.1007/s00222-010-0283-6.

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31

Martel, Yvan, and Frank Merle. "Asymptotic stability of solitons of the gKdV equations with general nonlinearity." Mathematische Annalen 341, no. 2 (December 15, 2007): 391–427. http://dx.doi.org/10.1007/s00208-007-0194-z.

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32

Bronski, Jared C., Mathew A. Johnson, and Todd Kapitula. "An index theorem for the stability of periodic travelling waves of Korteweg–de Vries type." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 141, no. 6 (November 15, 2011): 1141–73. http://dx.doi.org/10.1017/s0308210510001216.

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We consider the stability of periodic travelling-wave solutions to a generalized Korteweg–de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L2-critical KdV-4 equation that arises in the study of blow-up and the KdV-½ equation, which is an idealized model for plasmas.

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33

Martel, Yvan, and Frank Merle. "On the Nonexistence of Pure Multi-solitons for the Quartic gKdV Equation." International Mathematics Research Notices 2015, no. 3 (October 16, 2013): 688–739. http://dx.doi.org/10.1093/imrn/rnt214.

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34

Vinh, Nguyễn Tiến. "Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation." Nonlinearity 30, no. 12 (November 16, 2017): 4614–48. http://dx.doi.org/10.1088/1361-6544/aa8cab.

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35

Zhang, Zhi Fei. "Global well-posedness for gKdV-3 in Sobolev spaces of negative index." Acta Mathematica Sinica, English Series 24, no. 5 (May 2008): 857–66. http://dx.doi.org/10.1007/s10114-007-5597-y.

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36

Côte, Raphaël. "Construction of solutions to the subcritical gKdV equations with a given asymptotical behavior." Journal of Functional Analysis 241, no. 1 (December 2006): 143–211. http://dx.doi.org/10.1016/j.jfa.2006.04.007.

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37

Dias, João-Paulo, Mário Figueira, and Filipe Oliveira. "Well-posedness and existence of bound states for a coupled Schrödinger-gKdV system." Nonlinear Analysis: Theory, Methods & Applications 73, no. 8 (October 2010): 2686–98. http://dx.doi.org/10.1016/j.na.2010.06.049.

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38

Lan, Yang. "On continuation properties after blow-up time for $L^2$-critical gKdV equations." Revista Matemática Iberoamericana 36, no. 4 (January 7, 2020): 957–84. http://dx.doi.org/10.4171/rmi/1154.

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39

Yong, Zhu, and Dai Shiqiang. "On head-on collison between two gKdV solitary waves in a stratified fluid." Acta Mechanica Sinica 7, no. 4 (November 1991): 300–308. http://dx.doi.org/10.1007/bf02486737.

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40

Li, Zi-Liang. "Periodic Wave Solutions of a Generalized KdV-mKdV Equation with Higher-Order Nonlinear Terms." Zeitschrift für Naturforschung A 65, no. 8-9 (September 1, 2010): 649–57. http://dx.doi.org/10.1515/zna-2010-8-905.

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The Jacobin doubly periodic wave solution, the Weierstrass elliptic function solution, the bell-type solitary wave solution, the kink-type solitary wave solution, the algebraic solitary wave solution, and the triangular solution of a generalized Korteweg-de Vries-modified Korteweg-de Vries equation (GKdV-mKdV) with higher-order nonlinear terms are obtained by a generalized subsidiary ordinary differential equation method (Gsub-ODE method for short). The key ideas of the Gsub-ODE method are that the periodic wave solutions of a complicated nonlinear wave equation can be constructed by means of the solutions of some simple and solvable ODE which are called Gsub-ODE with higherorder nonlinear terms

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41

Zhu, Quanyong, Jinxi Fei, and Zhengyi Ma. "Residual Symmetry Analysis for Novel Localized Excitations of a (2+1)-Dimensional General Korteweg-de Vries System." Zeitschrift für Naturforschung A 72, no. 9 (August 28, 2017): 795–804. http://dx.doi.org/10.1515/zna-2017-0124.

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AbstractThe nonlocal residual symmetry of a (2+1)-dimensional general Korteweg-de Vries (GKdV) system is derived by the truncated Painlevé analysis. The nonlocal residual symmetry is then localized to a Lie point symmetry by introducing auxiliary-dependent variables. By using Lie’s first theorem, the finite transformation is obtained for the localized residual symmetry. Furthermore, multiple Bäcklund transformations are also obtained from the Lie point symmetry approach via the localization of the linear superpositions of multiple residual symmetries. As a result, various localized structures, such as dromion lattice, multiple-soliton solutions, and interaction solutions can be obtained through it; and these localized structures are illustrated by graphs.

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42

Arora, R., and A. kumar. "Soliton Solution of GKDV, RLW, GEW and GRLW Equations by Sine-hyperbolic Function Method." American Journal of Computational and Applied Mathematics 1, no. 1 (August 31, 2012): 1–4. http://dx.doi.org/10.5923/j.ajcam.20110101.01.

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43

Combet, Vianney, and Yvan Martel. "Sharp asymptotics for the minimal mass blow up solution of the critical gKdV equation." Bulletin des Sciences Mathématiques 141, no. 2 (March 2017): 20–103. http://dx.doi.org/10.1016/j.bulsci.2017.01.001.

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44

Gomes, Andressa, and Ademir Pastor. "Solitary wave solutions and global well-posedness for a coupled system of gKdV equations." Journal of Evolution Equations 21, no. 2 (April 2, 2021): 2167–93. http://dx.doi.org/10.1007/s00028-021-00676-4.

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45

Martel, Yvan, and Frank Merle. "Nonexistence of blow-up solution with minimal L 2 -mass for the critical gKdV equation." Duke Mathematical Journal 115, no. 2 (November 2002): 385–408. http://dx.doi.org/10.1215/s0012-7094-02-11526-9.

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46

Bhattarai, Santosh, Adán J. Corcho, and Mahendra Panthee. "Well-Posedness for Multicomponent Schrödinger–gKdV Systems and Stability of Solitary Waves with Prescribed Mass." Journal of Dynamics and Differential Equations 30, no. 2 (April 11, 2018): 845–81. http://dx.doi.org/10.1007/s10884-018-9660-4.

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47

LAI, X., Q. CAO, and E. TWIZELL. "The global domain of attraction and the initial value problems of a kind of GKdV equations." Chaos, Solitons & Fractals 23, no. 5 (March 2005): 1613–28. http://dx.doi.org/10.1016/s0960-0779(04)00413-8.

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48

Helal, M. A., and M. S. Mehanna. "A comparative study between two different methods for solving the general Korteweg–de Vries equation (GKdV)." Chaos, Solitons & Fractals 33, no. 3 (August 2007): 725–39. http://dx.doi.org/10.1016/j.chaos.2006.11.011.

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49

Sepúlveda, M. "Stability properties of a higher order scheme for a GKdV-4 equation modelling surface water waves." Calcolo 49, no. 4 (March 3, 2012): 269–91. http://dx.doi.org/10.1007/s10092-012-0055-3.

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50

AUDIARD, CORENTIN. "DISPERSIVE SCHEMES FOR THE CRITICAL KORTEWEG–DE VRIES EQUATION." Mathematical Models and Methods in Applied Sciences 23, no. 14 (October 10, 2013): 2603–46. http://dx.doi.org/10.1142/s0218202513500413.

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In this paper we study semi-discrete finite difference schemes for the critical Korteweg–de Vries equation (cKdV, which is gKdV for k = 4). We prove that the solutions of the discretized equation (using a two grid algorithm) satisfy dispersive estimates uniformly with respect to the discretization parameter. This implies convergence in a weak sense of the discrete solutions to the solution of the Cauchy problem even for rough L2(ℝ) initial data. We also prove a scattering result for the discrete equation, and show that the discrete scattering function converges to the continuous one. Finally rates of convergence are obtained for the approximation of a semi-linear equation with initial data in Hs, s > 0, yet a similar result remains open for the quasilinear cKdV equation. Our analysis relies essentially on the discrete Fourier transform and standard harmonic analysis on the real line.

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