Atomic Solution for Certain Gardner Equation
Abstract
:1. Introduction
2. Main Result
- (1)
- , a non zero atom.
- (2)
- or are linearly dependent.
3. A Further Result
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Caraballo, T.; Guo, B.; Tuan, N.; Wang, R. Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains. Proc. R. Soc. Edinb. Sect. A 2021, 151, 1700–1730. [Google Scholar] [CrossRef]
- Fan, E.G. Integrable Systems and Computer Algebra; Science Press: Beijing, China, 2004. [Google Scholar]
- Fang, J.P.; Zheng, C.L. New exact excitations and soliton fission and fusion for the (2 + 1)-dimensional Broer-Kaup-Kupershmidt system. Chin. Phys. 2005, 14, 669. [Google Scholar] [CrossRef]
- Ma, H.C.; Ge, D.J.; Yu, Y.D. New periodic wave solutions, localized excitations and their interaction for (2 + 1)-dimensional Burgers equation. Chin. Phys. B 2008, 17, 43–44. [Google Scholar] [CrossRef]
- Malfliet, W. Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 1992, 60, 650–654. [Google Scholar] [CrossRef]
- Wang, R. Long-time dynamics of stochastic lattice plate equations with nonlinear noise and damping. J. Dynam. Differ. Equ. 2021, 33, 767–803. [Google Scholar] [CrossRef]
- Wang, M. Solitary wave solutions for variant Boussinesq equations. Phys. Lett. A 1995, 199, 169–172. [Google Scholar] [CrossRef]
- Wang, R.; Wang, B. Random dynamics of p-Laplacian lattice systems driven by infinite-dimensional nonlinear noise. Stoch. Process. Appl. 2020, 130, 7431–7462. [Google Scholar] [CrossRef]
- Chen, P.; Wang, B.; Wang, R.; Zhang, X. Multivalued random dynamics of Benjamin-Bona-Mahony equations driven by nonlinear colored noise on unbounded domains. Math. Ann. 2022, 31. [Google Scholar] [CrossRef]
- Hereman, W.; Nuseir, A. Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math. Comput. Simul. 1997, 43, 13–27. [Google Scholar] [CrossRef]
- Hirota, R. The Direct Method in Soliton Theory; Cambridge University Press: Cambridge, UK, 2004; Volume 155. [Google Scholar]
- Ying, J.P.; Lou, S.Y. Multilinear variable separation approach in (3 + 1)-dimensions: The Burgers equation. Chin. Phys. Lett. 2003, 20, 1448. [Google Scholar] [CrossRef]
- Tang, X.Y.; Liang, Z.F. Variable separation solutions for the (3 + 1)-dimensional Jimbo-Miwa equation. Phys. Lett. A 2006, 351, 398–402. [Google Scholar] [CrossRef]
- Zhang, S.L.; Lou, S.Y.; Qu, C.Z. The Derivative-dependent functional variable separation for the evolution equations. Chin. Phys. 2006, 15, 2765–2776. [Google Scholar] [CrossRef]
- Liu, S.; Fu, Z.; Liu, S.; Zhao, Q. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 2001, 289, 69–74. [Google Scholar] [CrossRef]
- Ziqan, A.; Alhorani, M.; Khalil, R. Tensor product technique and the degenerate homogeneous abstract Cauchy problem. J. Appl. Funct. Anal. 2010, 5, 121–138. [Google Scholar]
- Ziqan, A.; Alhorani, M.; Khalil, R. Tensor product technique and nonhomogeneous degenerate Abstract Cauchy problem. Int. J. Appl. Math. Res. 2010, 23, 137–158. [Google Scholar]
- Al-Sharif, S.; Aljarrah, A.; Almefleh, H. Exact Solution of Burger’s Equation Using Tensor Product Technique. Eur. J. Pure Appl. Math. 2022, 15, 1444–1454. [Google Scholar] [CrossRef]
- Bekraoui, F.; Alhorani, M.; Khalil, R. Atomic solution of fractional abstract Cauchy problem of high order in Banach spaces. Eur. J. Pure Appl. Math. 2022, 15, 106–125. [Google Scholar] [CrossRef]
- Benkemache, I.; Alhorani, M.; Khalil, R. Tensor product and certain solutions of fractional wave type equation. Eur. J. Pure Appl. Math. 2021, 14, 942–948. [Google Scholar] [CrossRef]
- Khalil, R.; Al-Sharif, S.; Khamis, S. Second order abstract Cauchy problem of conformable ractional type. Int. J. Nonlinear Anal. Appl. 2022, 13, 1143–1150. [Google Scholar]
- Khamis, S.; Alhorani, M.; Khalil, R. Rank two solutions of the abstract Cauchy problem. J. Semigroup Theory Appl. 2018, 2018, 3. [Google Scholar]
- Seddiki, F.; Alhorani, M.; Khalil, R. Finite rank solution for conformable fractional degenerate first order abstract Cauchy problem in Hilbert spaces. Eur. J. Pure Appl. Math. 2021, 14, 493–505. [Google Scholar] [CrossRef]
- Khalil, R.; Abdullah, L. Atomic solution of certain inverse problems. Eur. J. Pure Appl. Math. 2010, 3, 725–729. [Google Scholar]
- Seddiki, F.; Alhorani, M.; Khalil, R. Tensor Product and Inverse Fractional Abstract Cauchy Problem. Rend. Del Circ. Mat. Palermo Ser. 2 2022, 12. [Google Scholar] [CrossRef]
- Al-Rab’a, A.; Al-Sharif, S.; Al-Khaleel, M. Double Conformable Sumudu Transform. Symmetry 2022, 14, 2249. [Google Scholar] [CrossRef]
- Wu, S.-L.; Al-Khaleel, M. Convergence analysis of the Neumann–Neumann waveform relaxation method for time-fractional RC circuits. Simul. Model. Pract. Theory 2016, 64, 43–56. [Google Scholar] [CrossRef]
- Wang, R.; Guo, B.; Wang, B. Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on RNdriven by nonlinear noise. Sci. China Math. 2021, 64, 2395–2436. [Google Scholar] [CrossRef]
- Wang, R.; Li, Y.; Wang, B. Random dynamics of fractional nonclassical diffusion equations driven by colored noise. Discrete Contin. Dyn. Syst. 2019, 39, 4091–4126. [Google Scholar] [CrossRef]
- Wang, R.; Shi, L.; Wang, B. Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on RN. Nonlinearity 2019, 32, 4524–4556. [Google Scholar] [CrossRef]
- Kilbas, A.; Srivastava, H.; Trujillo, J. Theory and Applications of Fractional Differential Equations, Math. Studies 204; North-Holland: New York, NY, USA, 2006. [Google Scholar]
- Samko, G.; Kilbas, A.; Marichev, A. Fractional Integrals and Derivatives, Theory and Applications; Gordon and Breach: Yverdon, Switzerland, 1993. [Google Scholar]
- Khalil, R.; Alhorani, M.; Yousef, A.; Sababheh, M. A new definition of fractional derivative. J. Comput. Appl. Math. 2014, 264, 65–70. [Google Scholar] [CrossRef]
- Al-Sharif, S.; Malkawi, A. Modification of conformable fractional derivative with classical properties. Ital. J. P. Appl. Math. 2020, 44, 30–39. [Google Scholar]
- Light, W.; Cheney, E. Approximation Theory in Tensor Product Spaces, Lecture Notes in Mathematics, 1169; Springer: Berlin, Germany; New York, NY, USA, 1985. [Google Scholar]
- Williams, D.P. Tensor products with bounded continuous functions. N. Y. J. Math. 2003, 9, 69–77. [Google Scholar]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Al-Khaleel, M.; Al-Sharif, S.; AlJarrah, A. Atomic Solution for Certain Gardner Equation. Symmetry 2023, 15, 440. https://doi.org/10.3390/sym15020440
Al-Khaleel M, Al-Sharif S, AlJarrah A. Atomic Solution for Certain Gardner Equation. Symmetry. 2023; 15(2):440. https://doi.org/10.3390/sym15020440
Chicago/Turabian StyleAl-Khaleel, Mohammad, Sharifa Al-Sharif, and Ameerah AlJarrah. 2023. "Atomic Solution for Certain Gardner Equation" Symmetry 15, no. 2: 440. https://doi.org/10.3390/sym15020440
APA StyleAl-Khaleel, M., Al-Sharif, S., & AlJarrah, A. (2023). Atomic Solution for Certain Gardner Equation. Symmetry, 15(2), 440. https://doi.org/10.3390/sym15020440