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Mathematics & Statistics

From Rogue Waves to Lump Waves

Rogue waves were once regarded as incredible as sea monsters and ghost ships; occasional reports of a ship that somehow survived a hundred foot wave were put down to hyperbole and grog. After all, the standard statistical models predicted that such tall waves would be incredibly rare — especially out at sea. But then, on 1 January 1995, rogue wave hit the Draupner E platform off Norway:

It was 85 feet high.

Since then, satellites have confirmed that rogue waves are a common phenomenon, and indeed, Professor Nail Akhmediev of Australian National University estimates that at any given time, there are about ten rogue waves somewhere out there.

Outside of human curiosity, rogue waves pose a continuing threat to human life and property, so the theory needs to catch up with empirical science. We begin at the beginning. Oceanic rogue waves are surface gravity waves with wave heights much larger than expected (according to the standard Gaussian model); by surface gravity wave, we mean that it moves on the surface, unlike sound waves that move through water. Oceanographers used to think that wave heights would be normally distributed — which would mean that very tall rogue waves would be extremely rare.

Waves were classically understood as manifestations of various wave equations, the most famous of which is \[ d^2/dx^2 + x = 0, \] whose solutions were the sine and cosine functions: That is what waves looked like: sine or cosine waves. This wave equation is linear in that we only add terms; we don't multiply them.

Recently, mathematicians have been using nonlinear differential equations to model a rogue wave as a kind of solitary wave (or soliton). One of these, the Kadomtsev-Petviashvili (KP) equation, has a solution that looks like this “lump wave”: This lump wave and higher-order lump waves involve rational functions, i.e., quotients of polynomials. And the above lump wave is special: the denominator is a quadratic polynomial.

Now for a technical problem: if we have a solution to a differential equation, might it misbehave in ways that have nothing to do with rogue waves and everything to do with the model? (Mathematicians spend a lot of time on these technical issues because ... oceanographers and meteorologists like to have models that don't misbehave.) One collection of technical problems would be resolved if the denominator — which we remember is a quadratic polynomial - is always positive.

USF Professor Wen-Xiu Ma and USF alumnus Yuan Zhou determined when a quadratic function of several variables, expressed with a single vector variable, could be positive, as follows. The coefficients were matrices and vectors, like this: if the function \(f\) is of three variables \(x\), \(y\), \(z\), then we can have a single vector variable \(v=[x,y,z]\) and a matrix \(A\) and two vectors \(\mathbf{b}\) and \(\mathbf{c}\) such that \(f=v^TAv-2b^Tv+c\), where \(v^T\) is the matrix \[ \begin{bmatrix} x \\ y \\ z \end{bmatrix}. \] They found conditions on \(A\), \(\mathbf{b}\), and \(\mathbf{c}\) equivalent to \(f\) being positive everywhere.

Wen-Xiu Ma Yuan Zhou

(Their paper is posted online.)

There is still a long way to go in testing these differential equations as descriptions of rogue waves. For example, suppose that a solution for a more sophisticated (and hopefully more accurate) model is a rational function whose denominator is not quadratic: we still do not have a good criterion for when non-quadratic polynomials are always positive.

Meanwhile, somewhere, out there, right now, ten rogue waves roam the seven seas...