Nonlinear self-adjointness through differential substitutions |
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| Institution: | 1. Institute of Electronics and Mechanical Engineering, Yuri Gagarin State Technical University of Saratov, Polytechnicheskaya 77, Saratov 410054, Russia;2. Department of Nonlinear Processes, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia;1. Research Center of Analysis and Control for Complex Systems, Chongqing University of Posts and Telecommunications, Chongqing 400065, China;2. Key Laboratory of Industrial Internet of Things & Networked Control, Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China;3. Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, China;1. A.I. Alikhanyan National Science Laboratory, 0036 Yerevan, Armenia;2. Departamento de Ciencias Exatas, Universidade Federal de Lavras, CP 3037, 37200-000 Lavras-MG, Brazil;3. Dipartimento di Scienza e Alta Tecnologia, Universitá degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy;4. I.N.F.N. Sezione di Milano, Via Celoria 16, 20133 Milano, Italy;5. Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR CNRS 6303, Université de Bourgogne, 21078 Dijon Cedex, France;6. Institute for Physical Research, 0203 Ashtarak-2, Armenia |
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| Abstract: | It is known (Ibragimov, 2011; Galiakberova and Ibragimov, 2013) 14,18] that the property of nonlinear self-adjointness allows to associate conservation laws of the equations under study, with their symmetries. In this paper we show that, even when the equation is nonlinearly self-adjoint with a non differential substitution, finding the explicit form of the differential substitution can provide new conservation laws associated to its symmetries. By using the general theorem on conservation laws (Ibragimov, 2007) 11] and the property of nonlinear self-adjointness we find some new conservation laws for the modified Harry-Dym equation. By using a differential substitution we construct a conservation law for the Harry-Dym equation, which has not been derived before using Ibragimov method. |
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| Keywords: | Self-adjointness Quasi self-adjointness Weak self-adjointness Nonlinear self-adjointness Symmetries Partial differential equations Conservation laws |
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