# Soliton solution for a diffusive system [closed]

With a simple model for bacterial diffusion, I get this partial derivative equation : $$\frac{\partial n}{\partial t} = D\frac{\partial^2 n}{\partial x^2} + d_1 n -d_2 n^2$$

where $n(x,t)$ is the population of bacteria, $d_1$ the rate of proliferation and $d_2 n$ is the rate of death for a bacteria.

I know that there should exist some soliton solution going from the solution $n=0$ to the solution $n=\frac{d_1}{d_2}$ but I have no clue how to treat such a problem. What are some methods to get about those soliton solutions?

## closed as off-topic by ACuriousMind♦, CuriousOne, Bernhard, Kyle Kanos, Rob JeffriesMay 25 '15 at 9:55

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At least for an equilibrium situation, where $\dfrac{\partial n}{\partial t}=0$ and $\dfrac{\partial n}{\partial x}=0$, you would easily see that

$$d_1 n - d_2 n^2=0,$$

with the two solutions you anticipated.

I am not sure what you mean with a soliton solution for an equation like these, as the basis of your equation is not a wave equation. Probably you mean solutions that also satisfy the equation:

$$\frac{\partial n}{\partial t} = D\frac{\partial^2 n}{\partial x^2}$$

But it would lead to the same result. I hope this is at least a partially satisfying answer.

• I'm actually looking for a solution (soliton= like $n(x,t) = f(x-ct)$ avec $f(u \rightarrow -\infty) =d_1/d_2$ and $f(u \rightarrow +\infty) =0$ even if it's a diffusion equation. Physically, the information I think is intersting is to find out how evolve a border between the two solution, what it's lengthscale and so on – sailx May 24 '15 at 14:19
• @sailx Ok, I understand your question better now. You may want to look into the error function, as it might help you. I am however a bit afraid for the quadratic term in the equation. – Bernhard May 24 '15 at 14:30
• Also, this may be more appropriate on the math stackexchange. – Bernhard May 24 '15 at 14:31