Heuristic Methods for Solving Large Scale Network Routing Problems: The Telpaking Problem

The inverse problem of the calculus of variations, i.e., the existence and formulation of variational principles for systems of nonlinear partial differential equations, is treated by application of a theorem due to Vainberg. Three types of principles are discussed: potential, alternate potential, and composite principles. Potential principles are those for which the equations, as written, admit a variational formulation. Alternate potential principles are those for which the equations admit a variational formulation only after a differential transformation of variables. Composite principles are those in which, in addition to the original variables, a set of adjoint variables is defined. A composite principle may always be formulated. Consistency relations for the existence of a potential are formulated for general systems of nonlinear partial differential equations of any order. These conditions are quite stringent and severely restrict the class of operators which admit a variational formulation. The results of the development are used to treat many well-known equations in nonlinear wave mechanics. These include the Burgers, Korteweg-deVries, and sine-Gordon equations and evolution and model equations for interfacial waves. The Navier-Stokes equations and the Boussinesq equations for an infinite Prandtl number fluid are briefly treated; these latter equations do not possess a potential principle.

[1]  L. E. Scriven,et al.  On the search for variational principles , 1967 .

[2]  G. Whitham,et al.  Non-linear dispersive waves , 1965, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[3]  D. J. Benney,et al.  The Evolution of Multi-Phase Modes for Nonlinear Dispersive Waves , 1970 .

[4]  J. C. Luke.  A perturbation method for nonlinear dispersive wave problems , 1966, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[5]  A. Scott,et al.  The soliton: A new concept in applied science , 1973 .

[6]  M. Ablowitz.  Applications of Slowly Varying Nonlinear Dispersive Wave Theories , 1971 .

[7]  S. M. Deshpande.  Connection between elliptic equations of navier-stokes type and theory of harmonic functionals , 1973 .

[8]  G. B. Whitham,et al.  Non-linear dispersion of water waves , 1967, Journal of Fluid Mechanics.

[9]  Shu-Kwan Chan.  Infinite Prandtl Number Turbulent Convection , 1971 .

[10]  R. L. Seliger,et al.  Variational principles in continuum mechanics , 1968, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[11]  F. H. Busse,et al.  Bounds for turbulent shear flow , 1970, Journal of Fluid Mechanics.

[12]  B. Finlayson.  Existence of Variational Principles for the Navier‐Stokes Equation , 1972 .

[13]  L. Howard,et al.  Bounds on Flow Quantities , 1972 .

[14]  M. Ablowitz.  Approximate Methods for Obtaining Multi‐Phase Modes in Nonlinear Dispersive Wave Problems , 1972 .

[15]  G. B. Whitham,et al.  Variational methods and applications to water waves , 1967, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[16]  Clark B. Millikan,et al.  LXXVII. On the steady motion of viscous, incompressible fluids; with particular reference to a variation principle , 1929 .

[17]  G. B. Whitham,et al.  Two-timing, variational principles and waves , 1970, Journal of Fluid Mechanics.

[18]  R. Atherton,et al.  ON THE DERIVATION OF EVOLUTION EQUATIONS FOR INTERFACIAL WAVES , 1976 .

[19]  G. Whitham.  A general approach to linear and non-linear dispersive waves using a Lagrangian , 1965, Journal of Fluid Mechanics.