Abstract
In this paper, we use a variational approach to show the analyticity of solitons (traveling waves of finite energy) and the existence of x-periodic traveling waves for a certain sixth order dispersive equation that arises in the study of the deformations of a hyperelastic compressible plate. We also establish that x-periodic traveling waves have almost the same shape of solitons as the period tends to infinity, by showing that a special sequence of periodic traveling wave solutions parameterized by the period converges to a soliton in a appropriate sense.
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References
Angulo, J.: Stability of cnoidal waves to Hirota-Satsuma systems. Matemática contemporánea 27, 189–223 (2004)
Angulo, J., Quintero, J.R.: Existence and orbital stability of cnoidal waves for a 1D Boussinesq equation. Int. J. Math. Math. Sci. 2007, 1–36 (2007)
Benjamin, T., Bona, J., Mahony, J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. A 272, 47–78 (1972)
Besov, O.V., Il’in, V.P., Nikolskii, S.M.: Integral Representations of Functions and Imbedding Theorems, vol. I. J. Wiley, New York (1978)
Bona, J., Liu, Y., Tom, M.: The Cauchy problem and stability of solitary-wave solutions for RLW-KP-type equations. J. Differ. Equ. 185, 437–482 (2002)
Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)
Chen, R.M.: Some nonlinear dispersive waves arising in compressible hyperelastic plates. Int. J. Eng. Sci. 4, 1188–1204 (2006)
Chen, R.M.: The Cauchy problem and the stability of solitary waves of a hyperelastic dispersive equation. Indiana U. J. Math. 57, 2949–2979 (2008)
Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 109–145 (1984)
Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 223–283 (1984)
Pankov, A.A., Pfl\(\ddot{\text{u}}\)ger, K.: Periodic and solitary traveling wave solutions for the generalized Kadomtsev–Petviashvili equation. Math. Methods Appl. Sci. 22(9), 733–752 (1999)
Quintero, J.R.: Solitons and periodic travelling waves for the 2D- generalized Benney–Luke equation. J. Appl. Anal. 9–10, 859–890 (2007)
Quintero, J.R.: From periodic travelling waves to solitons for a 2D water wave system. Methods Appl. Anal. 21(2), 241–264 (2014)
Soriano, F.: On the Cauchy problem for a K.P.-Boussinesq-Type system. J. Differ. Equ. 183, 79–107 (2002)
Saut, J., Tzvetkov, N.: Global well-posedness for the KP-BBM equations. AMRX 1, 1–16 (2004)
Willem, M.: Minimax Methods, Progress in Nonlinear Differential Equations and Their Applications. Birkh\(\ddot{\text{ a }}\)user, Boston (1996)
Zhanping, L., Jiabao, S.: Existence of solitary waves to a generalized Kadomtsev–Petviashvili. Acta Mathematica Scientia 32B(3), 1149–1156 (2012)
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A. M. Montes was supported by la Universidad del Cauca (Colombia) under the project No 5101.
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Montes, A.M. Solitons and Periodic Traveling Waves for a Hyperelastic Dispersive Equation. J Dyn Diff Equat 35, 2013–2033 (2023). https://doi.org/10.1007/s10884-022-10141-6
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DOI: https://doi.org/10.1007/s10884-022-10141-6