[**NONLINEAR SELF-ADJOINTNESS IN\
CONSTRUCTING CONSERVATION LAWS**]{}\
Nail H. Ibragimov\
Department of Mathematics and Science, Blekinge Institute of Technology,\
371 79 Karlskrona, Sweden

**Abstract.** The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the previous notions of self-adjoint [@ibr06a] and quasi self-adjoint [@ibr07d] nonlinear equations. The class of nonlinearly self-adjoint equations includes, in particular, all linear equations. Conservation laws associated with symmetries can be constructed for all nonlinearly self-adjoint differential equations and systems. The number of equations in systems can be different from the number of dependent variables.\
*Keywords*: Nonlinear self-adjointness, Strict and quasi self-adjointness, Conservation laws, Kompaneets equation, Reaction-diffusion model, Short pulse equation, Chaplygin gas, Approximate self-adjointness, Approximate conservation laws.\
MSC: 70S10, 35C99, 35G20\
PACS: 02.30.Jr, 11.15.-j, 02.20.Tw\

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  © 2010 N.H. Ibragimov.
  First published in *Archives of ALGA*, vol. 7/8, 2010-2011, pp. 1-99.
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[Part 1]{}\
Nonlinear self-adjointness {#part-1-nonlinear-self-adjointness .unnumbered}
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Preliminaries {#sa:eveq.int}
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The concept of self-adjointness of nonlinear equations was introduced [@ibr06a; @ibr07a] for constructing conservation laws associated with symmetries of differential equations. To extend the possibilities of the new method for constructing conservation laws the notion of quasi self-adjointness was suggested 