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arXiv 1107.4877math-phmath.MP
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非线性自伴性与守恒律

Nonlinear self-adjointness and conservation laws

Nail H. Ibragimov

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摘要：

引入了微分方程非线性自伴性的一般概念。它包括线性自伴性作为一种特殊情况。此外，它还包括作者前面介绍的严格自伴性和准自伴性。证明了具有非线性自伴性的方程可以通过适当的乘子等价地写成严格自伴形式。所有线性方程都具有非线性自伴性，因此可以改写为非线性严格自伴方程。例如，热方程$u_t - Δu = 0$在乘以$u^{-1}.$之后成为严格自伴的。对于所有具有非线性自伴性质的微分方程和系统，可以构造与对称性相关的守恒律。

英文摘要：

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 strict self-adjointness and quasi self-adjointness introduced earlier by the author. It is shown that the equations possessing the nonlinear self-adjointness can be written equivalently in a strictly self-adjoint form by using appropriate multipliers. All linear equations possess the property of nonlinear self-adjointness, and hence can be rewritten in a nonlinear strictly self-adjoint. For example, the heat equation $u_t - Δu = 0$ becomes strictly self-adjoint after multiplying by $u^{-1}.$ Conservation laws associated with symmetries can be constructed for all differential equations and systems having the property of nonlinear self-adjointness.

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