Second order partial derivatives

Diffusion equation

An example in which the spatial derivatives are of second order is the one-dimensional scalar diffusion equation, which describes heat flow and random walks:

$$\frac{\partial }{\partial t} A(x,t) = \kappa \frac{\partial ^2}{\partial x^2} A(x,t).$$ (2.9)

In this case, there is no potential term of form $\mbf{U}[\mbf{A}]$, and the partial differential operator is the one-dimensional scalar diffusion operator:

$$\mathcal{D} = \kappa \frac{\partial ^2}{\partial x^2}.$$ (2.10)

Initial value problems of this type are not the only problems possible with partial differential equations. We can also have boundary value problems, as in electrostatics; but we restrict ourselves to initial value problems in this lab. This equation can be solved completely analytically using Fourier transform methods, which we discuss later. A possible solution on the infinite line with vanishing boundary conditions at infinity is the Gaussian solution of form

$$A(x,t) = \frac{1}{\sigma (t)\sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma^2(t)}\right),$$ (2.11)

where $\sigma(t) = \sqrt{2 \kappa t + \sigma^2_0}$, and $\sigma_0$ is the initial Gaussian standard deviation.

Schrödinger equation

The Schrödinger equation is one of the most fundamental equations of quantum mechanics, and describes the complex probability amplitude $\Psi(t,x)$ of a particle-wave, as a function of space and time. In one space dimension, it has the form:

$$i\hbar \frac{\partial }{\partial t} \psi(x,t) = -\frac{\hbar^2}{2m} \frac{\partial ^2}{\partial x^2} \psi(x,t) + V(x) \psi(x,t).$$ (2.12)

In this case, there is a derivative term of the form:

$$\mathcal{D} = \frac{i\hbar}{2m} \frac{\partial ^2}{\partial x^2} \psi(x,t),$$ (2.13)

together with a potential term of form $\mbf{U}[\psi](\mbf{x},t) = V(x) \psi(x,t)/(i\hbar)$, which is linear in the field $\psi(x,t)$ itself. This equation has a rich family of possible solutions, including the well-known plane-wave solutions for a particle of known momentum $p = \hbar k = m v$, and energy $E = \hbar \omega = V_0 + m v^2/2$, moving in a uniform potential of $V(x) = V_0$:

$$\psi(x,t) = \psi_0 e^{i (kx - \omega t)}.$$ (2.14)

More generally, write:

$$\frac{\partial }{\partial t} \psi(x,t) = i \kappa \frac{\partial ^2}{\partial x^2} \psi(x,t) + U(x) \psi(x,t),$$ (2.15)

then the solution for a Gaussian wave-packet (with $V(x) = 0$) is obtained just by replacing $\kappa$ with $i\kappa$ in the solution to the diffusion equation, with $\kappa =\hbar/2 m$, i.e.,

$$\psi(x,t) = \frac{1}{\sigma(t) \sqrt{2\pi}} \exp\left( ikx - \frac{x^2}{2\sigma^2(t)} \right)$$ (2.16)

where $\sigma(t) = \sqrt{2\kappa i t + \sigma^2_0}$, and $\sigma_0$ is the initial Gaussian standard deviation.

Nonlinear Schrödinger equation

In this case, there is a derivative term as above, together with a potential term of form $\mbf{U}[\Psi(t,\mbf{x})] = i \gamma \psi(x,t) \vert\psi(x,t)\vert^2$, which is nonlinear in the field $\psi(x,t)$. This is an example of a nonlinear partial differential equation. Such equations are very important, because most real-life PDEs are actually nonlinear to some extent, and the commonly used linear equations are only an approximation to this true situation.

For example, the nonlinear Schrödinger equation describes solitons in optical fiber pulse propagation, which is linear to a first approximation at low intensities, but becomes measurably nonlinear at higher intensities:

$$\frac{\partial }{\partial t} \psi(x,t) = i \kappa \frac{\partial ^2}{\partial x^2} \psi(x,t) + i \gamma \psi(x,t) \vert\psi(x,t)\vert^2.$$ (2.17)

Note that, if you compare this case with the above example, the potential term has the opposite sign; i.e. this equation corresponds to an attractive potential which increases with the modulus of the field. The physical result is that a stable, nonlinear wave can form, which creates its own stabilizing potential that moves with the wave itself. This is called a soliton.

Burger's equation

The full Burger's equation, including viscosity, has an equation of form:

$$\frac{\partial }{\partial t} A(x,t) = -v \frac{\partial }{\partial x} [A - A^2/A_0] + \kappa \frac{\partial ^2 A}{\partial x^2}.$$ (2.18)

These equations are sometimes used to describe hydrodynamics or traffic flow. This equation models the fact that cars travel more slowly when the traffic density increases, and it includes a viscosity factor which models the finite time-scale of a reaction to changes in conditions--or, in the fluid case, the effects of fluid friction. The extra diffusion term not only makes the equation more realistic, it also makes the equation easier to treat numerically, by removing some of the numerical instabilities present in the inviscid Burgers equation.