Modular Hopf equation

Abstract

In this Letter we study modular Hopf equation of the form ut+|u|ux=0and obtain explicit form of its exact solution in the Fourier space, for the particular initial conditions of a sine wave. This solution exists for a finite time before wave breaking. We also demonstrate the qualitative difference between the Fourier spectra of the modular and classical Hopf equations. In contrast to the classical Hopf equation, which generates one second harmonic during the initial stage of time evolution, in the modular Hopf equation all even harmonics are absent, but there exist infinitely many odd Fourier modes.