stw : Continuation of Sharp Traveling Waves.

This demo illustrates the computation of sharp traveling wave front solutions to nonlinear diffusion problems of the form

$$w_t = A(w) w_{xx} + B(w) w_x^{2} + C(w),$$

with $A(w) = a_1 w + a_2 w^{2}$, $B(w) = b_0 + b_1 w + b_2 w^{2}$, and $C(w) = c_0 + c_1 w + c_2 w^{2}$. Such equations can have sharp traveling wave fronts as solutions, i.e., solutions of the form $w(x,t)=u(x+ct)$ for which there is a $z_0$ such that $u(z)=0$ for $z \ge z_0$, $u(z) \not= 0$ for $z < z_0$, and $u(z) \rightarrow constant$ as $z \rightarrow -\infty$. These solutions are actually generalized solutions, since they need not be differentiable at $z_0$.

Specifically, in this demo a homotopy path will be computed from an analytically known exact sharp traveling wave solution of

$$w_t = 2w w_{xx} + 2 w_x^{2} + w(1-w), \leqno(1)$$

to a corresponding sharp traveling wave of

$$w_t = (2w+w^{2}) w_{xx} + w w_x^{2} + w(1-w). \leqno(2)$$

This problem is also considered in DoKeKe:91b DoKeKe:91b. For these two special cases the functions $A,B,C$ are defined by the coefficients in Table 18.3.

Table 18.3: Problem coefficients in demo stw.

	$a_1$	$a_2$	$b_0$	$b_1$	$b_2$	$c_0$	$c_1$	$c_2$
Case (1)	2	0	2	0	0	0	1	-1
Case (2)	2	1	0	1	0	0	1	-1

With $w(x,t)=u(x+ct)$, $z=x+ct$, one obtains the reduced system

$$\begin{array}{cl} & u_1'(z) = u_2, \\ & u_2'(z) = \bigl[c u_2 - B(u_1) u_2^{2} - C(u_1) \bigr]/A(u_1). \\ \end{array}$$ (18.5)

To remove the singularity when $u_1=0$, we apply a nonlinear transformation of the independent variable (see Ar:80 Ar:80), viz., ${d / d \tilde z} = A(u_1) {d / dz}$, which changes the above equation into

$$\begin{array}{cl} & u_1'(\tilde z) = A(u_1) u_2, \\ & u_2'(\tilde z) = c u_2 - B(u_1) u_2^{2} - C(u_1). \\ \end{array}$$ (18.6)

Sharp traveling waves then correspond to heteroclinic connections in this transformed system.

Finally, we map $[0,T] \rightarrow [0,1]$ by the transformation $\xi = \tilde z / T$. With this scaling of the independent variable, the reduced system becomes

$$\begin{array}{cl} & u_1'(\xi) = T A(u_1) u_2, \\ & u_2'(\xi) = T \bigl[ c u_2 - B(u_1) u_2^{2} - C(u_1)\bigr]. \\ \end{array}$$ (18.7)

For Case 1 this equation has a known exact solution, namely,

$$u(\xi) = \frac{1 }{ 1 + exp(T\xi) }, \qquad v(\xi) = \frac{ -\frac{1 }{ 2} }{ 1 + exp(-T\xi) }.$$

This solution has wave speed $c=1$. In the limit as $T \rightarrow \infty$ its phase plane trajectory connects the stationary points $(1,0)$ and $(0,-\frac{1 }{ 2})$.

The sharp traveling wave in Case 2 can now be obtained using the following homotopy. Let $(a_1,a_2,b_0,b_1,b_2) = (1-\lambda) (2,0,2,0,0) + \lambda (2,1,0,1,0)$. Then as $\lambda$ varies continuously from 0 to 1, the parameters $(a_1,a_2,b_0,b_1,b_2)$ vary continously from the values for Case 1 to the values for Case 2.

Table 18.4: Commands for running demo stw.

AUTO -COMMAND	ACTION
! mkdir stw	create an empty work directory
cd stw	change directory
demo('stw')	copy the demo files to the work directory
run(c='stw.1')	continuation of the sharp traveling wave
sv('stw')	save output-files as b.stw, s.stw, d.stw