q-Advanced Models for Tsunami and Rogue Waves

Abstract EN

A wavelet Kq(t), that satisfies the q-advanced differential equation Kq′(t)=Kq(qt) for q>1, is used to model N-wave oscillations observed in tsunamis.

Although q-advanced ODEs may seem nonphysical, we present an application that model tsunamis, in particular the Japanese tsunami of March 11, 2011, by utilizing a one-dimensional wave equation that is forced by Fq(t,x)=Kq(t)qSin(x).

The profile Fq is similar to tsunami models in present use.

The function Sin (t) q is a wavelet that satisfies a q-advanced harmonic oscillator equation.

It is also shown that another wavelet, Cos (t) q , matches a rogue-wave profile.

This is explained in terms of a resonance wherein two small amplitude forcing waves eventually lead to a large amplitude rogue.

Since wavelets are used in the detection of tsunamis and rogues, the signal-analysis performance of Kq and Cos q is examined on actual data.