Volume 12, no. 2Pages 37 - 46

On a Nonlinear Problem of the Breaking Water Waves

M. Kirane, B.T. Torebek
The paper is devoted to the initial boundary value problem for the Korteweg-de Vries-Benjamin-Bona-Mahony equation in a finite domain. This particular problem arises from the phenomenon of long wave with small amplitude in fluid. For certain initial-boundary problems for the Korteweg-de Vries-Benjamin-Bona-Mahony equation, we obtain the conditions of blowing-up of global and travelling wave solutions in finite time. The proof of the results is based on the nonlinear capacity method. In closing, we provide the exact and numerical examples.
Full text
breaking waves; Korteweg-de Vries-Benjamin-Bona-Mahony equation; blow-up of solution; initial-boundary problems.
1. Sarpkaya T., Isaacson M. Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold, 1981. DOI: 10.1115/1.3162189
2. Benjamin T.B., Bona J.L., Mahony J.J. Model Equations for Long Waves in Nonlinear Dispersive Systems. Philosophical Transactions of the Royal Society of London, 1972, vol. 272, pp. 47-78. DOI: 10.1098/rsta.1972.0032
3. Francius M., Pelinovsky E.N., Slunyaev A.V. Wave Dynamics in Nonlinear Media with Two Dispersionless Limits for Long and Short Waves. Physics Letters, 2001, vol. 280, no. 2, pp. 53-57. DOI: 10.1016/S0375-9601(01)00042-1
4. Jie Li, Kangsheng Liu. Well-Posedness of Korteweg-de Vries-Benjamin-Bona-Mahony Equation on a Finite Domain. Journal of Mathematical Analysis and Applications, 2017, vol. 452, no. 1, pp. 611-633. DOI: 10.1016/j.jmaa.2017.02.038
5. Korpusov M.O., Yushkov E.V. Local Solvability and Blow-Up for Benjamin-Bona-Mahony-Burgers, Rosenau-Burgers and Korteweg-de Vries-Benjamin-Bona-Mahony Equations. Electronic Journal of Differential Equations, 2014, vol. 69, pp. 1-16.
6. Pokhozhaev S.I. On the Singular Solutions of the Korteweg-de Vries Equation. Mathematical Notes, 2010, vol. 88, no. 5, pp. 741-747. DOI: 10.1134/S0001434610110131
7. Pokhozhaev S.I. On the Nonexistence of Global Solutions for Some Initial-Boundary Value Problems for the Korteweg-de Vries Equation. Differential Equations, 2011, vol. 47, no. 4, pp. 488-493. DOI: 10.1134/S0012266111040045
8. Pokhozhaev S.I. On the Nonexistence of Global Solutions of the Cauchy Problem for the Korteweg-de Vries Equation. Functional Analysis and Its Applications, 2012, vol. 46, no. 4, pp. 279-286. DOI: 10.1007/s10688-012-0035-z
9. Pokhozhaev S.I. Blow-Up of Smooth Solutions of the Korteweg-de Vries Equation. Nonlinear Analysis: Theory, Methods and Applications, 2012, vol. 75, no. 12, pp. 4688-4698. DOI: 10.1016/j.na.2011.08.021
10. Pokhozhaev S.I. Essentially Nonlinear Capacities Induced by Differential Operators. Proceedings of the USSR Academy of Sciences, 1997, vol. 357, no. 5, pp. 592-594.
11. Mitidieri E., Pokhozhaev S.I. A Priori Estimates and Blow-Up of Solutions of Nonlinear Partial Differential Equations and Inequalities. Proceedings of the Steklov Institute of Mathematics, 2001, vol. 234, pp. 1-362.
12. Mitidieri E., Pokhozhaev S.I. Towards a Unified Approach to Nonexistence of Solutions for a Class of Differential Inequalities. Milan Journal of Mathematics, 2004, vol. 72, pp. 129-162. DOI: 10.1007/s00032-004-0032-7
13. Martel Y., Merle F. Blow Up in Finite Time and Dynamics of Blow Up Solutions for the L^2-сritical Generalized KDV Equation. Journal of the American Mathematical Society, 2002, vol. 15, no. 3, pp. 617-664.
14. Martel Y., Merle F. Stability of Blow-Up Profile and Lower Bounds for Blow-Up Rate for the Critical Generalized KDV Equation. Annals of Mathematics, 2014, vol. 155, no. 1, pp. 235-280. DOI: 10.2307/3062156
15. Martel Y., Merle F., Rapha'el P. Blow Up for the Critical Generalized Korteweg-de Vries Equation. I: Dynamics Near the Solition. Acta Mathematica, 2014, vol. 212, no. 1, pp. 59-140. DOI: 10.1007/s11511-014-0109-2