The four-dimensional Martínez Alonso-Shabat equation: nonlinear self-adjointness and conservation laws. (English) Zbl 1361.35014
Summary: We show that the four-dimensional Martínez Alonso-Shabat equation is nonlinearly self-adjoint with differential substitution and the required differential substitution is just the admitted adjoint symmetry and vice versa. By means of computer algebra system, we obtain a number of local and nonlocal symmetries admitted by the equations under study. Then such symmetries are used to construct conservation laws of the equation under study and its reductions.
MSC:
| 35A30 | Geometric theory, characteristics, transformations in context of PDEs |
| 70S10 | Symmetries and conservation laws in mechanics of particles and systems |
| 35B06 | Symmetries, invariants, etc. in context of PDEs |
| 68W30 | Symbolic computation and algebraic computation |
References:
| [1] | MorozovOI. The four‐dimensional Martínez Alonso-Shabat equation: differential coverings and recursion operators. Journal of Geometry and Physics. 2014; 85:75-80. · Zbl 1301.37044 |
| [2] | MorozovOI, SergyeyevA. The four‐dimensional Martínez Alonso-Shabat equation: reductions and nonlocal symmetries. Journal of Geometry and Physics. 2014; 85:40-45. · Zbl 1301.37045 |
| [3] | Martínez AlonsoL, ShabatAB. Hydrodynamic reductions and solutions of a universal hierarchy. Theoretical & Mathematical Physics. 2004; 104:1073-1085. · Zbl 1178.37067 |
| [4] | SergyeyevA. Infinite hierarchies of nonlocal symmetries of the Chen-Kontsevich-Schwarz type for the oriented associativity equations. Journal of Physics A: Mathematical and Theoretical. 2009; 42:404017 (15pp). · Zbl 1186.37036 |
| [5] | Krasil’shchikJ, VerbovetskyA. Geometry of jet spaces and integrable systems. Journal of Geometry and Physics. 2011; 61:1633-1674. · Zbl 1230.58005 |
| [6] | OlverPJ. Applications of Lie Groups to Differential Equationations. Springer‐Verlag: New York, 1993. · Zbl 0785.58003 |
| [7] | BlumanGW, CheviakovAF, AncoSC. Applications of Symmetry Methods to Partial Differential Equationations. Springer‐Verlag: New York, 2010. · Zbl 1223.35001 |
| [8] | KaraAH, MahomedFM. Noether‐type symmetries and conservation laws via partial Lagrangians. Nonlinear Dynamics. 2006; 5:367-383. · Zbl 1121.70014 |
| [9] | NazR, MahomedFM, MasonDP. Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics. Applied Mathematics and Computation. 2008; 205:212-230. · Zbl 1153.76051 |
| [10] | PopovychRO, SergyeyevA. Conservation laws and normal forms of evolution equations. Physics Letters A. 2010; 374:2210-2217. · Zbl 1237.35142 |
| [11] | IbragimovNH. Nonlinear self‐adjointness in constructing conservation laws, Arch. ALGA. 2011; 7/8:1-99. |
| [12] | IbragimovNH. Quasi self‐adjoint differential equations, Arch. ALGA. 2007; 4:55-60. |
| [13] | GandariasML. Weak self‐adjoint differential equations. Journal of Physics A: Mathematical and Theoretical. 2011; 44:262001 (6pp). · Zbl 1223.35203 |
| [14] | IbragimovNH, AvdoninaED. Nonlinear self‐adjointness conservation, laws and the construction of solutions of partial differential equations using conservation laws. Russian Mathematics Surveys. 2013; 68:889-921. · Zbl 1286.35013 |
| [15] | IbragimovNH. A new conservation theorem. Journal of Mathematical Analysis and Applications. 2007; 333:311-328. · Zbl 1160.35008 |
| [16] | KrasilshchikIS, VinogradovAM. Nonlocal symmetries and the theory of coverings: an addendum to Vinogradov’s local symmetries and conservation laws. Acta Applicandae Mathematicae. 1984; 2:79-96. · Zbl 0547.58043 |
| [17] | GandariasML. Nonlinear self‐adjointness through differential substitutions. Communications in Nonlinear Science and Numerical Simulation. 2014; 19:3523-3528. · Zbl 1470.35014 |
| [18] | ZhangZY. On the existence of conservation law multiplier for partial differential equations. Communications in Nonlinear Science and Numerical Simulation. 2015; 20:338-351. · Zbl 1310.35017 |
| [19] | ChaoluT, BlumanGW. An algorithmic method for showing existence of nontrivial non‐classical symmetries of partial differential equations without solving determining equations. Journal of Mathematical Analysis and Applications. 2014; 411:281-296. · Zbl 1308.35013 |
| [20] | ChaoluT. An algorithm for the complete symmetry classification of differential equations based on Wu’s method. Journal of Engineering Mathematics. 2010; 66:181-199. · Zbl 1204.35021 |
| [21] | MarvanM. 1996. Another look on recursion operators. In Differential Geometry and Applications. Masaryk University: Brno; 393-402. · Zbl 0870.58106 |
| [22] | MarvanM, SergyeyevA. Recursion operator for the stationary Nizhnik-Veselov-Novikov equation. Journal of Physics A: Mathematical and General. 2003; 36:L87-L92. · Zbl 1039.37055 |
| [23] | MarvanM, SergyeyevA. Recursion operators for dispersionless integrable systems in any dimension. Inverse Problems. 2012; 28:025011 (12pp). · Zbl 1234.35314 |
| [24] | SergyeyevA. Recursion Operators for Multidimensional Integrable Systems,. 2015; arXiv:1501.01955. |
| [25] | BlaszakM. Classical R‐matrices on Poisson algebras and related dispersionless systems. Physics Letters A. 2002; 297:191-195. · Zbl 0995.37059 |
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