This dissertation revolves around various mathematical aspects of nonlinear wave motion
in viscoelasticity and free surface flows.
The introduction is devoted to the physical derivation of the stressstrain constitutive
relations from the first principles of Newtonian mechanics and is accessible to a broad
audience. This derivation is not necessary for the analysis carried out in the rest of the
thesis, however, is very useful to connect the differentlooking partial differential equations
(PDEs) investigated in each subsequent chapter.
In the second chapter we investigate a multidimensional scalar wave equation with
memory for the motion of a viscoelastic material described by the most general linear
constitutive law between the stress, strain and their rates of change. The model equation is
rewritten as a system of firstorder linear PDEs with relaxation and the wellposedness of
the Cauchy problem is established.
In the third chapter we consider the Euler equations describing the evolution of a perfect,
incompressible, irrotational fluid with a free surface. We focus on the Hamiltonian
description of surface waves and obtain a recursion relation which allows to expand the
Hamiltonian in powers of wave steepness valid to arbitrary order and in any dimension. In
the case of pure gravity waves in a twodimensional flow there exists a symplectic coordinate
transformation that eliminates all cubic terms and puts the Hamiltonian in a Birkhoff
normal form up to order four due to the unexpected cancellation of the coefficients of all
fourth order nongeneric resonant terms. We explain how to obtain higherorder vanishing
coefficients.
Finally, using the properties of the expansion kernels we derive a set of nonlinear evolution
equations for unidirectional gravity waves propagating on the surface of an ideal fluid
of infinite depth and show that they admit an exact traveling wave solution expressed in
terms of Lambertâ€™s Wfunction. The only other known deep fluid surface waves are the
Gerstner and Stokes waves, with the former being exact but rotational whereas the latter
being approximate and irrotational. Our results yield a wave that is both exact and irrotational,
however, unlike Gerstner and Stokes waves, it is complexvalued.
Date of Award  Jul 24 2020 

Original language  English (US) 

Awarding Institution   Computer, Electrical and Mathematical Sciences and Engineering


Supervisor  Peter Markowich (Supervisor) 

 weakly nonlinear surface waves
 Birkhoff normal form
 Integrable systems
 Hamiltonian PDEs
 Water waves problem
 Zener model