Authors

Long Ju, Jian Zhou, Yufeng Zhang

Abstract

This article mainly uses two methods of solving the conservation laws of two partial differential equations and a system of equations. The first method is to construct the conservation law directly and the second method is to apply the Ibragimov method to solve the conservation laws of the target equation systems, which are constructed based on the symmetric rows of the target equation system. In this paper, we select two equations and an equation system, and we try to apply these two methods to the combined KdV-MKdV equation, the Klein-Gordon equation and the generalized coupled KdV equation, and simply verify them. The combined KdV-MKdV equation describes the wave propagation of bound particles, sound waves and thermal pulses. The Klein-Gordon equation describes the nonlinear sine-KG equation that simulates the motion of the Josephson junction, the rigid pendulum connected to the stretched wire, and the dislocations in the crystal. And the coupled KdV equation has also attracted a lot of research due to its importance in theoretical physics and many scientific applications. In the last part of the article, we try to briefly analyze the Hamiltonian structures and adjoint symmetries of the target equations, and calculate their linear soliton solutions.

Citation

  • Journal: Communications in Analysis and Mechanics
  • Year: 2023
  • Volume: 15
  • Issue: 2
  • Pages: 24–49
  • Publisher: American Institute of Mathematical Sciences (AIMS)
  • DOI: 10.3934/cam.2023002

BibTeX

@article{Ju_2023,
  title={{Conservation laws analysis of nonlinear partial differential equations and their linear soliton solutions and Hamiltonian structures}},
  volume={15},
  ISSN={2836-3310},
  DOI={10.3934/cam.2023002},
  number={2},
  journal={Communications in Analysis and Mechanics},
  publisher={American Institute of Mathematical Sciences (AIMS)},
  author={Ju, Long and Zhou, Jian and Zhang, Yufeng},
  year={2023},
  pages={24--49}
}

Download the bib file

References

  • Olver PJ (1990) Symmetries and Differential Equations (G. W. Bluman and S. Kumei). SIAM Rev 32(3):517–519. https://doi.org/10.1137/1032114 – 10.1137/1032114
  • Tu G-zhang (1989) The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. Journal of Mathematical Physics 30(2):330–338. https://doi.org/10.1063/1.528449 – 10.1063/1.528449
  • P. J. Olver, Applications of Lie Groups to Differential Equations, Springer Science and Business Media New York, NY, (2012). https://doi.org/10.1016/0001-8708(88)90053-9
  • San S, Akbulut A, Ünsal Ö, Taşcan F (2016) Conservation laws and double reduction of (2+1) dimensional Calogero–Bogoyavlenskii–Schiff equation. Math Methods in App Sciences 40(5):1703–1710. https://doi.org/10.1002/mma.4091 – 10.1002/mma.4091
  • Simkins L (1963) Instructions as discriminative stimuli in verbal conditioning and awareness. The Journal of Abnormal and Social Psychology 66(3):213–219. https://doi.org/10.1037/h0047923 – 10.1037/h0047923
  • ANCO SC, BLUMAN G (2002) Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications. Eur J Appl Math 13(5):545–566. https://doi.org/10.1017/s095679250100465x – 10.1017/s095679250100465x
  • ANCO SC, BLUMAN G (2002) Direct construction method for conservation laws of partial differential equations Part II: General treatment. Eur J Appl Math 13(5):567–585. https://doi.org/10.1017/s0956792501004661 – 10.1017/s0956792501004661
  • Anco SC, Bluman G (1997) Direct Construction of Conservation Laws from Field Equations. Phys Rev Lett 78(15):2869–2873. https://doi.org/10.1103/physrevlett.78.2869 – 10.1103/physrevlett.78.2869
  • Anco SC, Bluman G (1998) Integrating factors and first integrals for ordinary differential equations. Eur J Appl Math 9(3):245–259. https://doi.org/10.1017/s0956792598003477 – 10.1017/s0956792598003477
  • Ibragimov NH (2011) Nonlinear self-adjointness and conservation laws. J Phys A: Math Theor 44(43):432002. https://doi.org/10.1088/1751-8113/44/43/432002 – 10.1088/1751-8113/44/43/432002
  • Wang H, Zhang Y (2021) Self-adjointness and conservation laws of Burgers-type equations. Mod Phys Lett B 35(09):2150161. https://doi.org/10.1142/s021798492150161x – 10.1142/s021798492150161x
  • Ibragimov NH (2006) Integrating factors, adjoint equations and Lagrangians. Journal of Mathematical Analysis and Applications 318(2):742–757. https://doi.org/10.1016/j.jmaa.2005.11.012 – 10.1016/j.jmaa.2005.11.012
  • Ibragimov NH (2007) A new conservation theorem. Journal of Mathematical Analysis and Applications 333(1):311–328. https://doi.org/10.1016/j.jmaa.2006.10.078 – 10.1016/j.jmaa.2006.10.078
  • Ibragimov NH (2018) Conservation laws and non-invariant solutions of anisotropic wave equations with a source. Nonlinear Analysis: Real World Applications 40:82–94. https://doi.org/10.1016/j.nonrwa.2017.08.005 – 10.1016/j.nonrwa.2017.08.005
  • Anco SC, Wang B (2021) A formula for symmetry recursion operators from non-variational symmetries of partial differential equations. Lett Math Phys 111(3):70. https://doi.org/10.1007/s11005-021-01413-1 – 10.1007/s11005-021-01413-1
  • Anco SC (2016) Symmetry properties of conservation laws. Int J Mod Phys B 30(28n29):1640003. https://doi.org/10.1142/s0217979216400038 – 10.1142/s0217979216400038
  • Gu X, Ma W (2018) On a class of coupled Hamiltonian operators and their integrable hierarchies with two potentials. Math Methods in App Sciences 41(10):3779–3789. https://doi.org/10.1002/mma.4864 – 10.1002/mma.4864
  • Manukure S (2018) Finite-dimensional Liouville integrable Hamiltonian systems generated from Lax pairs of a bi-Hamiltonian soliton hierarchy by symmetry constraints. Communications in Nonlinear Science and Numerical Simulation 57:125–135. https://doi.org/10.1016/j.cnsns.2017.09.016 – 10.1016/j.cnsns.2017.09.016
  • Zhang J, Gongye Y, Ma W (2022) The relationship between the conservation laws and multi‐Hamiltonian structures of the Kundu equation. Math Methods in App Sciences 45(16):9006–9020. https://doi.org/10.1002/mma.8288 – 10.1002/mma.8288
  • Anco SC, Gandarias ML, Recio E (2018) Conservation Laws, Symmetries, and Line Soliton Solutions of Generalized KP and Boussinesq Equations with p-Power Nonlinearities in Two Dimensions. Theor Math Phys 197(1):1393–1411. https://doi.org/10.1134/s004057791810001x – 10.1134/s004057791810001x
  • A.P. Marquez, M.L. Gandarias, S.C. Anco, Conservation laws, symmetries, and line solitons of a Kawahara-KP equation, arXiv preprint arXiv, preprint, arXiv: 2211.03904.
  • Chen C, Jiang Y-L (2017) Lie Group Analysis, Exact Solutions and New Conservation Laws for Combined KdV–mKdV Equation. Differ Equ Dyn Syst 28(4):827–840. https://doi.org/10.1007/s12591-017-0351-0 – 10.1007/s12591-017-0351-0
  • Ak T, Karakoc SBG, Biswas A (2017) Application of Petrov-Galerkin finite element method to shallow water waves model: Modified Korteweg-de Vries equation. Scientia Iranica 24(3):1148–1159. https://doi.org/10.24200/sci.2017.4096 – 10.24200/sci.2017.4096
  • Gazi Karakoc SB (2018) A Quartic Subdomain Finite Element Method for the Modified KdV Equation. Stat, optim inf comput 6(4). https://doi.org/10.19139/soic.v6i4.485 – 10.19139/soic.v6i4.485
  • Seydi Battal Gazi Karakoc (2018) Numerical solutions of the modified KdV Equation with collocation method. Malaya J Mat 6(04):835–842. https://doi.org/10.26637/mjm0604/0020 – 10.26637/mjm0604/0020
  • Ak T, Karakoc SBG, Biswas A (2017) A New Approach for Numerical Solution of Modified Korteweg-de Vries Equation. Iran J Sci Technol Trans Sci 41(4):1109–1121. https://doi.org/10.1007/s40995-017-0238-5 – 10.1007/s40995-017-0238-5
  • Mohammadizadeh F, Rashidi S, Hejazi SR (2021) Space–time fractional Klein-Gordon equation: Symmetry analysis, conservation laws and numerical approximations. Mathematics and Computers in Simulation 188:476–497. https://doi.org/10.1016/j.matcom.2021.04.015 – 10.1016/j.matcom.2021.04.015
  • Satsuma J, Hirota R (1982) A Coupled KdV Equation is One Case of the Four-Reduction of the KP Hierarchy. J Phys Soc Jpn 51(10):3390–3397. https://doi.org/10.1143/jpsj.51.3390 – 10.1143/jpsj.51.3390
  • Karakoc SBG, Saha A, Sucu D (2020) A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: Generalized Korteweg-de Vries equation. Chinese Journal of Physics 68:605–617. https://doi.org/10.1016/j.cjph.2020.10.010 – 10.1016/j.cjph.2020.10.010
  • Zhang Y, Han Z, Tam H-W (2013) An integrable hierarchy and Darboux transformations, bilinear Bäcklund transformations of a reduced equation. Applied Mathematics and Computation 219(11):5837–5848. https://doi.org/10.1016/j.amc.2012.11.086 – 10.1016/j.amc.2012.11.086
  • Ablowitz MA, Clarkson PA (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press – 10.1017/cbo9780511623998
  • Tam H-W, Zhang Y (2012) An integrable system and associated integrable models as well as Hamiltonian structures. Journal of Mathematical Physics 53(10):103508. https://doi.org/10.1063/1.4752721 – 10.1063/1.4752721
  • Mikhailov AV (1981) The reduction problem and the inverse scattering method. Physica D: Nonlinear Phenomena 3(1-2):73–117. https://doi.org/10.1016/0167-2789(81)90120-2 – 10.1016/0167-2789(81)90120-2
  • Zhang H-Y, Zhang Y-F (2022) Spectral analysis and long-time asymptotics of complex mKdV equation. Journal of Mathematical Physics 63(2):021509. https://doi.org/10.1063/5.0073909 – 10.1063/5.0073909
  • Wang H, Zhang Y (2020) Two Nonisospectral Integrable Hierarchies and its Integrable Coupling. Int J Theor Phys 59(8):2529–2539. https://doi.org/10.1007/s10773-020-04519-9 – 10.1007/s10773-020-04519-9
  • Gao XN, Lou SY, Tang XY (2013) Bosonization, singularity analysis, nonlocal symmetry reductions and exact solutions of supersymmetric KdV equation. J High Energ Phys 2013(5):29. https://doi.org/10.1007/jhep05(2013)029 – 10.1007/jhep05(2013)029
  • Hu X-R, Lou S-Y, Chen Y (2012) Explicit solutions from eigenfunction symmetry of the Korteweg–de Vries equation. Phys Rev E 85(5):056607. https://doi.org/10.1103/physreve.85.056607 – 10.1103/physreve.85.056607
  • Lou SY, Hu X, Chen Y (2012) Nonlocal symmetries related to Bäcklund transformation and their applications. J Phys A: Math Theor 45(15):155209. https://doi.org/10.1088/1751-8113/45/15/155209 – 10.1088/1751-8113/45/15/155209
  • Lou S-Y, Hu X-B (1997) Infinitely many Lax pairs and symmetry constraints of the KP equation. Journal of Mathematical Physics 38(12):6401–6427. https://doi.org/10.1063/1.532219 – 10.1063/1.532219
  • Gazizov RK, Ibragimov NH, Lukashchuk SY (2015) Nonlinear self-adjointness, conservation laws and exact solutions of time-fractional Kompaneets equations. Communications in Nonlinear Science and Numerical Simulation 23(1-3):153–163. https://doi.org/10.1016/j.cnsns.2014.11.010 – 10.1016/j.cnsns.2014.11.010
  • Chen C, Zhou J, Zhao S, Feng B (2022) Integrable Coupling of Expanded Isospectral and Non-Isospectral Dirac Hierarchy and Its Reduction. Symmetry 14(12):2489. https://doi.org/10.3390/sym14122489 – 10.3390/sym14122489