Hamilton-Jacobi Formalism

Singular perturbation analysis does not provide a fully analytical result for the very important case of KPP kinetics. It is not possible to go beyond the first order in 5, because the exact unperturbed solution is not known and the integrals in the solvability condition diverge. Proceeding as in Sect. 4.2.1 for (6.50) with KPP kinetics, we obtain to leading order the following equation for the action functional (e = 0)  

We carry out the calculations explicitly for two simple choices of where (6.80) has an exact solution. The first one is (x) = x as in the previous section. In this case, (6.80) yields, together with the boundary conditions. 

The exact expression for the velocity, after inverting the hyperbolic scaling, is given by 

The local velocity approach for KPP kinetics yields v = 2 r eXf). The differential equation for the front position is 

In Fig. 6.9 we compare (6.86) and (6.90) with the numerical solution of (6.50) for different values of . We observe in general good agreement after an initial transient. However, the front velocity (6.87) is in better agreement with numerical solutions 

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By introducing the quantum postulate huo = E — V, the energy in excess of a constant potential, and the velocity of a matter wavefront, in Hamilton-Jacobi formalism (Goldstein, 1980), c = JT/2m, these equations (in 3D) transform into the familiar set of Schrodinger equations ... 

The Hamilton-Jacobi formalism can be extended to incorporate initial condition without compact support by considering... 

Using the hyperbolic scaling and the Hamilton-Jacobi formalism show that the front... 

The Hamilton-Jacobi formalism, on the other hand, only holds for KPP kinetics, but in contrast to singular perturbation analysis there is no need to assume either weak or smooth heterogeneities. The local velocity approach is based on the assumption that for weak and smooth heterogeneities the velocity of the front is given by the local value of the reaction rate r and the diffusion coefficient D at each spatial point, i.e., the front velocity coincides with the instantaneous Fisher velocity V 2y/r x)D x). In general, this simple-minded approach is not consistent with results from the other analytical methods or with numerical solutions. 

Apply the Hamilton-Jacobi formalism to the O Shaughnessy-Procaccia equation (6.7) and show that the front velocity behaves like v ... 

We have assumed logistic population growth, with K being the carrying capacity of the environment. The probability of a newborn being a disperser is /u. = rj/(rj +r2>. Note that if rj 0, the whole population disperses and (7.17) becomes the standard RD equation. Since the kinetics satisfy the KPP criteria, we can calculate the front velocity from the Hamilton-Jacobi formalism. The corresponding Hamiltonian is... 

In this model, an invasive cancer, where the tumor edge advances as a propagating front into normal tissue, corresponds to a transition to a stable state containing tumor cells, state III or IV. The Hamilton-Jacobi formalism provides the propagation velocity of a front connecting state II to state HI, an invasive tumor front. 

Using the Hamilton-Jacobi formalism, we obtain the Hamilton-Jacobi equation... 

Hamilton formalism Hamilton-Jacobi formalism Hilbert space... 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

Hamilton-Jacobi equation, molecular systems, modulus-phase formalism, 262-265 Lagrangean density correction term, 270 nearly nonrelativistic limit, 269... 

This equation has been used by Sundstrom and coworkers [151] and adapted to the analysis of femtosecond spectral evolution as monitored by the bond-twisting events in barrierless isomerization in solution. The theoretical derivation of Aberg et al. establishes a link between the Smoluchowski equation with a sink and the Schrodinger equation of a solute coupled to a thermal bath. The reader is referred to this important work for further theoretical details and a thorough description of the experimental set up. It is sufficient to say here that the classical link is established via the Hamilton-Jacobi equation formalism. By using the standard ansatz Xn(X,t)= A(X,i)cxp(S(X,t)/i1l), where S(X,t) is the action of the dynamical system, and neglecting terms in once this... 

In order to obtain a more compact formulation of the mixed quantum-classical equations we use a Hamilton-Jacobi-like formalism for the propagation of the quantum degree of freedom as in earlier studies [23], A similar approach has been introduced by Nettesheim, Schiitte and coworkers [54, 55, 56], TTie formalism presented here is based on recent investigations of the present authors [23], This formalism can be summarized as follows. Starting from the Hamiltonian Eqn. (2.2) and averaging over the x- and y-mode, respectively, gives... 

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