Boussinesq Denklemlerinin Hirota Direct Metod ile Tam Çözümleri

Year 2024, Volume: 24 Issue: 05, 1113 - 1119

Halide Gümüş , Abdullah Baykal

Abstract

Boussinesq Denklemleri (BSQ) bu makalenin odak noktasıdır. İlk olarak, nonlineer evolüsyon denklemlere çoklu soliton çözümler oluşturmak için kullanılan Hirota'nın D operatörüne ilişkin temel bir genel bakış sunuyoruz. Daha sonra dördüncü dereceden BSQ ile ilgili bazı detaylar veriliyor ve bir soliton çözüm bulmak için Hirota Direct yöntemini kullanıyoruz. Hirota'nın bilineer yaklaşımı aynı zamanda nonlineer evolüsyon denklem olan altıncı dereceden Boussinesq benzeri denklem sınıfını çözmek için de kullanılır. Sonuçlar, bu yaklaşımın tam integre edilebilirlik gerektirdiğini doğrulamıştır.

Keywords

Hirota Direct Metod, Boussinesq Denklemleri, Soliton çözümler, mathematica12

Exact Solutions of Boussinesq Equations By Hirota Direct Method

Abstract

Boussinesq Equations (BSQ) are the focus of this article. First, we provide a basic overview of Hirota's D operator, which is used to build multi-soliton solutions for equations involving nonlinear evolution. After that, some details regarding fourth-order BSQ are provided, and we use Hirota's direct method to find a one-solution solution. Hirota's bilinear approach is also used to solve a class of sixth-order Boussinesq-like equations with nonlinear evolution. The outcomes verified that this approach requires complete integrability.

Keywords

Hirota's Direct Method, Boussinesq equations, Soliton solutions, mathematica 12

References

• Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H. 1973. Nonlinear evolution equations and inverse scattering. Physical Review Letters, 31(2), 125-127. https://doi.org/10.1103/PhysRevLett.31.125
• Ablowitz, M. J., Kaup, D. J., Newell, A. C., & Segur, H. 1974. The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems. Studies in Applied Mathematics, 53(4), 249-315. https://doi.org/10.1002/sapm1974534249
• Ablowitz, M.J., Segur, H. 1981. Solitons and the inverse scattering transform. SIAM Studies in Applied Mathematics, 425 pages, SIAM, Philadelphia, PA.
• Anderson, J. D., 2011. Computational Fluid Dynamics: The Basics with Applications, John D. Anderson (alan editörü), McGraw-Hill Education, New York. 1-121.
• Boussinesq, J. 1877. Théorie de l’écoulement tourbillonnant des liquides. Mémoires présentés par divers savants à l’Académie des Sciences,23, 1-680.
• Cushman-Roisin, B. 1994. Introduction to Geophysical Fluid Dynamics, Bernard Cushman-Roisin (alan editörü), Prentice-Hall, New Jersey.
• Daripa, P., Rajan, K., D., 2002. Analytical and Numerical Studies of a singularly perturbed Boussinesq equation, Applied Mathematics and Computation, 126, 1-30. https://doi.org/10.1016/S0096-3003(01)00166-7.
• Gardner, C.S., Greene, J. M., Kruskal, M.D., Miura, R.M. 1967. Physical Review Letters, 19(19):1095-1097. https://doi.org/10.1103/PhysRevLett.19.1095.
• Hirota, R. 1971. Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons. Phys. Rev. Lett,27,1192. https://doi.org/10.1103/PhysRevLett.27.1192.
• Hirota, R. 1972. Exact Solution of the Modified Korteweg-de Vries Equation for Multiple Collisions of Solitons. Journal of the Physical Society of Japan, 33(5):1456-1458. https://doi.org/10.1143/JPSJ.33.1456
• Hirota, R. 1973. Exact Envelope-Soliton Solutions of a Nonlinear Wave Equation. Journal of Mathematical Physics,14(7):805-809. https://doi.org/10.1063/1.1666399
• Hirota, R. 1973. Exact N-soliton solutions of the wave equation of long waves in shallow-water and in nonlinear lattices, Journal Mathematical Physics, 14: 810- 4. https://doi.org/10.1063/1.1666400
• Hirota, R. 1980. Direct methods in soliton thoery. Solitons. Topics in Current Physics, vol. 17. Berlin: Springer-Verlag. https://doi.org/10.1007/978-3-642-81448-8_5.
• Hirota, R. 2004. The Direct Method in Soliton Theory. New York: Cambridge University Press, 12-58.
• Kay,I., Moses, H.E. 1956. Reflectionless Transmission through Dielectrics and Scattering Potentials. Journal of Applied Physics. 27,1503-1508. https://doi.org/10.1063/1.1722296.
• Korteweg, D. J. 1895. On the Change of Form of Long Waves advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves. Philosophical Magazine, 39(5), 422-443.
• Lax, P. D. 1968. Integrals of Nonlinear Equations of Evolution and Solitary Waves. Communications on Pure and Applied Mathematics, 21(5), 467-490. https://doi.org/10.1002/cpa.3160210503.
• Matsuno, Y. 1984. Bilinear Transformation Method. Academic Press, New York.
• Nakamura, A. 1979. A Direct Method of Calculating Periodic Wave Solutions to Nonlinear Evolution Equations. I. Exact Two-Periodic Wave Solution. Journal of the Physical Society of Japan, 47, pp. 1701-1705. https://doi.org/10.1143/JPSJ.47.1701.
• Russell, J. S. 1844. Report on Waves. Report of the 14th Meeting of the British Association for the Advancement of Science, York, London, 311. https://doi.org/10.4236/jamp.2014.24003.
• Whitham, G.B. 1984. Comments on periodic waves and solitons. IMA Journal of Applied Mathematics, 32 ,353-366. https://doi.org/10.1093/imamat/32.1-3.353
• Zabusky, N., Kruskal,M.D. 1965. Physical Review Letters, 15,240. http://dx.doi.org/10.1103/PhysRevLett.15.240.
• Zakharov, V. E., Shabat, A. B. 1972. Exact Theory of Two-dimensional Self-focusing and One dimensional Self modulation of Wave in Nonlinear Media. Soviet Journal of Experimental and Theoretical Physics,34,62-69.