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A class of nonlinear wave equations in \(2+1\) , of the form ( 3.49 ), which is sufficiently general, and also capable of analysis is the following vector-valued wave equation, \(\displaystyle \Box \phi + \nabla _\phi V = 0 \quad \Box \mathbf {v} + \nabla _{\mathbf {v}}V = 0 \,, \) where \(\Box \) is the d’Alembertian \(\displaystyle \Box := \frac {\partial ^2\ }{\partial t^2} -\frac {\partial ^2\ }{\partial x^2} -\frac {\partial ^2\ }{\partial y^2}\,, \) and \(V(\phi ,\mathbf {v})\) is a given superquadratic potential.

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Example: Nonlinear Wave Equation in \(2+1\)

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  • 发表日期：

    2025年

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  • Timothy J. Burchell,
  • Thomas J. Bridges

摘要

A class of nonlinear wave equations in \(2+1\) , of the form ( 3.49 ), which is sufficiently general, and also capable of analysis is the following vector-valued wave equation, \(\displaystyle \Box \phi + \nabla _\phi V = 0 \quad \Box \mathbf {v} + \nabla _{\mathbf {v}}V = 0 \,, \) where \(\Box \) is the d’Alembertian \(\displaystyle \Box := \frac {\partial ^2\ }{\partial t^2} -\frac {\partial ^2\ }{\partial x^2} -\frac {\partial ^2\ }{\partial y^2}\,, \) and \(V(\phi ,\mathbf {v})\) is a given superquadratic potential.