Hopf-Cole transformation

A solution of Eq. 10.17 can be derived using the Cole-Hopf transform. This solution gives the elution profile of a finite width pulse at the end of an infinitely long... 

Applying the Cole-Hopf transformation p = (f)/o)liiJi, this phase dynamics equation is transformed to a simple form analogous to the equation... 

A straightforward integration of system (15.13) can be problematic due to the long transients. Another approach, which also applies for two dimensional lattices, relies on the Cole-Hopf transformation of system (15.13). Assume that the (pi are small so that it is possible to approximate the coupling function (15.14) around zero by F(0) (e " l) + 0 (p ). After... 

Eq. (15.25) is well defined since the components of q" do not change sign. Anderson localization theory [45] predicts exponentially decaying localized states for one and two dimensional lattices with some localization length I, which after applying the reverse Cole-Hopf transformation yields the observed tent-shape phase profile with wavelength A 7/. Concentric waves emerge when A becomes smaller than the system size. 

When y = 0 (W = 0), the surface tension term is removed and (6.10) reduces to Burger s equation. In this case, the Cole - Hopf transformation further reduces it to heat equation. Since a > 0 the Cole - Hopf transformation produces a heat equation backward in time and initial disturbance will then grow without limit. Hence, we shall include the surface tension term and discuss equation (6.10) when a > 0 and y > 0. The full KS equation (6.10) is capable of generating solutions in the form of irregularity fluctuating quasi-periodic waves. This KS model equation provides a mechanism for the saturation of an instability, in which the energy in long-wave instabilities is transferred to short-wave modes which are then damped by surface tension. 

Burger s equation can be solved exactly using the Cole-Hopf transformation ... 

Equation (3.3.5) represents a nonlinear phase diffusion equation. It is equivalent to the Burgers equation in the case of one space dimension (Chap. 6). It is known that the Burgers equation can be reduced to a linear diffusion equation through a transformation called the Hopf-Cole transformation (Burgers, 1974), and essentially the same is true for (3.3.5) in an arbitrary dimension. We shall take advantage of this fact in Chap. 6 when analytically discussing a certain form of chemical waves. 

This is essentially the same as the well-known Hopf-Cole transformation which reduces the Burgers equation to a simple diffusion equation (Burgers, 1974). The same transformation reduces (6.3.2) to the linear equation... 

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