Symbolic computation of conservation laws of nonlinear partial differential equations in multi‐dimensions
Abstract A direct method for the computation of polynomial conservation laws of polynomial systems of nonlinear partial differential equations (PDEs) in multi‐dimensions is presented. The method avoids advanced differential‐geometric tools. Instead, it is solely based on calculus, variational calculus, and linear algebra. Densities are constructed as linear combinations of scaling homogeneous terms with undetermined coefficients. The variational derivative (Euler operator) is used to compute the undetermined coefficients. The homotopy operator is used to compute the fluxes. The method is illustrated with nonlinear PDEs describing wave phenomena in fluid dynamics, plasma physics, and quantum physics. For PDEs with parameters, the method determines the conditions on the parameters so that a sequence of conserved densities might exist. The existence of a large number of conservation laws is a predictor of complete integrability. The method is algorithmic, applicable to a variety of PDEs,
Abstract A direct method for the computation of polynomial conservation laws of polynomial systems of nonlinear partial differential equations (PDEs) in multi‐dimensions is presented. The method avoids advanced differential‐geometric tools. Instead, it is solely based on calculus, variational calculus, and linear algebra. Densities are constructed as linear combinations of scaling homogeneous terms with undetermined coefficients. The variational derivative (Euler operator) is used to compute the undetermined coefficients. The homotopy operator is used to compute the fluxes. The method is illustrated with nonlinear PDEs describing wave phenomena in fluid dynamics, plasma physics, and quantum physics. For PDEs with parameters, the method determines the conditions on the parameters so that a sequence of conserved densities might exist. The existence of a large number of conservation laws is a predictor of complete integrability. The method is algorithmic, applicable to a variety of PDEs, and can be implemented in computer algebra systems such as Mathematica, Maple, and REDUCE. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006
Executive Summary
The article presents a direct method for computing polynomial conservation laws of nonlinear partial differential equations in multi-dimensions. This method avoids advanced differential-geometric tools and instead relies on calculus, variational calculus, and linear algebra. It constructs densities as linear combinations of scaling homogeneous terms with undetermined coefficients and uses the variational derivative and homotopy operator to compute these coefficients and fluxes. The method is illustrated with examples from fluid dynamics, plasma physics, and quantum physics, and can be implemented in computer algebra systems.
Key Points
- ▸ Direct method for computing polynomial conservation laws
- ▸ Avoids advanced differential-geometric tools
- ▸ Applicable to nonlinear PDEs in multi-dimensions
Merits
Algorithmic Approach
The method is algorithmic, making it suitable for implementation in computer algebra systems, which enhances its practical applicability.
Demerits
Limited to Polynomial Systems
The method is restricted to polynomial systems of nonlinear PDEs, which might limit its applicability to more complex or non-polynomial systems.
Expert Commentary
The article contributes significantly to the field of nonlinear PDEs by providing a straightforward and algorithmic method for computing conservation laws. This approach not only simplifies the computational process but also makes it more accessible, especially for those without extensive background in differential geometry. The method's applicability to various fields, including fluid dynamics and quantum physics, underscores its potential for interdisciplinary research and practical applications. However, its limitation to polynomial systems suggests a need for further research to extend its applicability to more complex systems.
Recommendations
- ✓ Further research to extend the method to non-polynomial systems
- ✓ Implementation of the method in a variety of computer algebra systems to enhance accessibility
