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1.
Wheeler, Miles H.
Large-Amplitude Solitary Water Waves with Vorticity and
Surface Pressure.
Degree: PhD, Mathematics, 2014, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:386210/ 
► The water wave equations describe the motion of an incompressible inviscid fluid under the influence of gravity which is bounded above by a free surface…
(more)
▼ The water wave equations describe the motion of an
incompressible inviscid fluid under the influence of gravity which
is bounded above by a free surface under constant (atmospheric)
pressure. In the first part of this thesis, we construct exact
traveling solitary waves of large amplitude with an arbitrary
distribution of vorticity. We use a degree-theoretic continuation
argument to construct a global connected set of symmetric solitary
waves of elevation, whose profiles decrease monotonically on either
side of a central crest. In the second part we construct
large-amplitude symmetric solitary waves generated by a
non-constant pressure on the free surface. This set includes waves
of depression whose profiles increase monotonically from a central
trough where the surface pressure is at its lowest, as well as
waves of elevation whose profiles decrease monotonically from a
central crest where the surface pressure is at its highest. There
may also be two waves in this connected set with identical surface
pressure, only one of which is a wave of depression.
Advisors/Committee Members: Strauss, Walter (Director), Guo, Yan (Reader), Sandstede, Björn (Reader).
Subjects/Keywords: vorticity
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APA (6th Edition):
Wheeler, M. H. (2014). Large-Amplitude Solitary Water Waves with Vorticity and
Surface Pressure. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:386210/
Chicago Manual of Style (16th Edition):
Wheeler, Miles H. “Large-Amplitude Solitary Water Waves with Vorticity and
Surface Pressure.” 2014. Doctoral Dissertation, Brown University. Accessed April 15, 2021.
https://repository.library.brown.edu/studio/item/bdr:386210/.
MLA Handbook (7th Edition):
Wheeler, Miles H. “Large-Amplitude Solitary Water Waves with Vorticity and
Surface Pressure.” 2014. Web. 15 Apr 2021.
Vancouver:
Wheeler MH. Large-Amplitude Solitary Water Waves with Vorticity and
Surface Pressure. [Internet] [Doctoral dissertation]. Brown University; 2014. [cited 2021 Apr 15].
Available from: https://repository.library.brown.edu/studio/item/bdr:386210/.
Council of Science Editors:
Wheeler MH. Large-Amplitude Solitary Water Waves with Vorticity and
Surface Pressure. [Doctoral Dissertation]. Brown University; 2014. Available from: https://repository.library.brown.edu/studio/item/bdr:386210/
2.
Hadzic, Mahir.
Stability and instability in the Stefan problem with surface
tension.
Degree: PhD, Applied Mathematics, 2010, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:11068/
► We develop a high-order nonlinear energy method to study the stability of steady states of the Stefan problem with surface tension. There are two prominent…
(more)
▼ We develop a high-order nonlinear energy method to
study the stability of steady states of the Stefan problem with
surface tension. There are two prominent classes of steady states:
flat planes and round spheres.In the case of steady planes, we
prove that the equilibria are always stable, asymptotically
converging to a nearby flat hypersurface, in arbitrary dimesnions.
Our proof relieson an energy method along the fixed domain.In the
case of steady spheres, we establish sharp nonlinear stability and
instability criterion in arbitrary dimensions. Our nonlinear
stability proof relies on an energy method along the moving domain,
and the discovery of a new ''momentum conservation law''. Our
nonlinear instability proof relies on a variational framework which
leads to the sharp growth rate estimate for the linearized problem,
as well as a bootstrap framework to overcome the nonlinear
perturbation with severe high-order derivatives.The instability
result is surprising, as it stands in stark contrast to the known
stability results for the related Mullins-Sekerka
problem.
Advisors/Committee Members: Guo, Yan (Director), Dafermos, Constantine (Reader), Strauss, Walter (Reader).
Subjects/Keywords: partial differential equations
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APA (6th Edition):
Hadzic, M. (2010). Stability and instability in the Stefan problem with surface
tension. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:11068/
Chicago Manual of Style (16th Edition):
Hadzic, Mahir. “Stability and instability in the Stefan problem with surface
tension.” 2010. Doctoral Dissertation, Brown University. Accessed April 15, 2021.
https://repository.library.brown.edu/studio/item/bdr:11068/.
MLA Handbook (7th Edition):
Hadzic, Mahir. “Stability and instability in the Stefan problem with surface
tension.” 2010. Web. 15 Apr 2021.
Vancouver:
Hadzic M. Stability and instability in the Stefan problem with surface
tension. [Internet] [Doctoral dissertation]. Brown University; 2010. [cited 2021 Apr 15].
Available from: https://repository.library.brown.edu/studio/item/bdr:11068/.
Council of Science Editors:
Hadzic M. Stability and instability in the Stefan problem with surface
tension. [Doctoral Dissertation]. Brown University; 2010. Available from: https://repository.library.brown.edu/studio/item/bdr:11068/
3.
Carter, Paul A.
Fast Pulses with Oscillatory Tails in the FitzHugh-Nagumo
System.
Degree: PhD, Mathematics, 2016, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:674229/
► The FitzHugh-Nagumo equations are known to admit fast traveling pulses that have monotone tails and arise as the concatenation of Nagumo fronts and backs in…
(more)
▼ The FitzHugh-Nagumo equations are known to admit fast
traveling pulses that have monotone tails and arise as the
concatenation of Nagumo fronts and backs in an appropriate singular
limit, where a parameter ε goes to zero. These pulses are
known to be nonlinearly stable with respect to the underlying PDE.
Numerical studies indicate that the FitzHugh-Nagumo system exhibits
stable traveling pulses with oscillatory tails. In this work, the
existence and stability of such pulses is proved analytically in
the singular perturbation limit near parameter values where the
FitzHugh-Nagumo system exhibits folds. The existence proof utilizes
geometric blow-up techniques combined with the exchange lemma: the
main challenge is to understand the passage near two fold points on
the slow manifold where normal hyperbolicity fails. For the
stability result, similar to the case of monotone tails, stability
is decided by the location of a nontrivial eigenvalue near the
origin of the PDE linearization about the traveling pulse. We prove
that this real eigenvalue is always negative. However, the
expression that governs the sign of this eigenvalue for oscillatory
pulses differs from that for monotone pulses, and we show indeed
that the nontrivial eigenvalue in the monotone case scales with
ε, while the relevant scaling in the oscillatory case is
ε
2/3. Finally a mechanism is proposed that explains the
transition from single to double pulses that was observed in
earlier numerical studies, and this transition is constructed
analytically using geometric singular perturbation theory and
blow-up techniques.
Advisors/Committee Members: Sandstede, Bjorn (Director), Holmer, Justin (Reader), Strauss, Walter (Reader).
Subjects/Keywords: dynamical systems
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APA (6th Edition):
Carter, P. A. (2016). Fast Pulses with Oscillatory Tails in the FitzHugh-Nagumo
System. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:674229/
Chicago Manual of Style (16th Edition):
Carter, Paul A. “Fast Pulses with Oscillatory Tails in the FitzHugh-Nagumo
System.” 2016. Doctoral Dissertation, Brown University. Accessed April 15, 2021.
https://repository.library.brown.edu/studio/item/bdr:674229/.
MLA Handbook (7th Edition):
Carter, Paul A. “Fast Pulses with Oscillatory Tails in the FitzHugh-Nagumo
System.” 2016. Web. 15 Apr 2021.
Vancouver:
Carter PA. Fast Pulses with Oscillatory Tails in the FitzHugh-Nagumo
System. [Internet] [Doctoral dissertation]. Brown University; 2016. [cited 2021 Apr 15].
Available from: https://repository.library.brown.edu/studio/item/bdr:674229/.
Council of Science Editors:
Carter PA. Fast Pulses with Oscillatory Tails in the FitzHugh-Nagumo
System. [Doctoral Dissertation]. Brown University; 2016. Available from: https://repository.library.brown.edu/studio/item/bdr:674229/
4.
Kim, Chanwoo.
Initial Boundary Value Problem of the Boltzmann
Equation.
Degree: PhD, Mathematics, 2011, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:11308/
► In this thesis, we study some boundary problems of the Boltzmann equation and the Boltzmann equation with the large external potential.If the gas is contained…
(more)
▼ In this thesis, we study some boundary problems of the
Boltzmann equation and the Boltzmann equation with the large
external potential.If the gas is contained in a bounded domain or
flows past a solid bodies, the Boltzmann equation must be
accompanied by boundary conditions, which describe the interaction
of the gas with the solid walls. The basic boundary conditions are
in-flow injection, diffuse reflection, bounce-back, and specular
reflection boundary conditions.First, we demonstrate that
discontinuity of the Boltzmann solution is created at the
non-convex part of the grazing boundary, then propagate only along
the forward characteristics inside the domain before it hits on the
boundary again with the in-flow injection, diffuse reflection and
bounce-back boundary conditions. The main ingredient is the new
regularizing effect of the gain term.Second, we prove the global
well-posedness of the Boltzmann equation with the specular
reflection boundary condition when the domain is the cylindrical
shape. The main ingredients are introducing collision-pair map to
show the transversality and developing the graph comparison method
to characterize grazing velocities for the non-convex domains.If
the gas particles are charged and there is background electonic
potential, the Boltzmann equation is accompanied by the field term
∇χΦ⋅∇ᵥF in the equation. The global well-posedness in the presence
of the large external potential is an important open problem. We
use Asano's work, showing the transversality in the presence of
potentials, in L
p- L
∞framework to resolve this problem.
Advisors/Committee Members: Guo, Yan (Director), Strauss, Walter (Reader), Holmer, Justin (Reader).
Subjects/Keywords: partial differential equation
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Manager
APA (6th Edition):
Kim, C. (2011). Initial Boundary Value Problem of the Boltzmann
Equation. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:11308/
Chicago Manual of Style (16th Edition):
Kim, Chanwoo. “Initial Boundary Value Problem of the Boltzmann
Equation.” 2011. Doctoral Dissertation, Brown University. Accessed April 15, 2021.
https://repository.library.brown.edu/studio/item/bdr:11308/.
MLA Handbook (7th Edition):
Kim, Chanwoo. “Initial Boundary Value Problem of the Boltzmann
Equation.” 2011. Web. 15 Apr 2021.
Vancouver:
Kim C. Initial Boundary Value Problem of the Boltzmann
Equation. [Internet] [Doctoral dissertation]. Brown University; 2011. [cited 2021 Apr 15].
Available from: https://repository.library.brown.edu/studio/item/bdr:11308/.
Council of Science Editors:
Kim C. Initial Boundary Value Problem of the Boltzmann
Equation. [Doctoral Dissertation]. Brown University; 2011. Available from: https://repository.library.brown.edu/studio/item/bdr:11308/
5.
Ben-Artzi, Jonathan.
Linear Instability of Nonmonotone Super Heated
Plasmas.
Degree: PhD, Mathematics, 2011, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:11412/
► In this work we prove spectral instability of certain types of equilibria of the relativistic Vlasov-Maxwell system of equations, which describes the evolution of a…
(more)
▼ In this work we prove spectral instability of certain
types of equilibria of the relativistic Vlasov-Maxwell system of
equations, which describes the evolution of a hot, sparse plasma,
where collisions between particles may be ignored.Making a purely
growing mode ansatz, with growth exponent λ, we linearize the
Vlasov equation about an equilibrium, where λ serves as a
parameter. Using the reversibility of the Vlasov equation, we
express the particle distribution as a function of the
electromagnetic potentials, and substitute this expression into
Maxwell's equations. Luckily, they turn out to be a selfadjoint
system
M
λ, depending upon the parameter λ. To this selfadjoint
problem we must find a solution, with 0 < λ < ∞.To find a
solution to the selfadjoint problem we use an idea of Z. Lin: we
let the parameter λ vary between 0 and ∞, and attempt to keep track
of the spectrum of
M
λ, hoping to find some intermediate value 0 < λ
0< ∞ for which
M
λ
0has a nontrivial kernel.The main difficulties lie in
the fact that
M
λhas an unbounded spectrum that extends to both -∞ and
+∞, and in the fact that as λ > 0,
M
λtends to its limit
M
0only strongly, but not in operator norm.We overcome
these difficulties by truncating the spectrum, as was first
suggested by Z. Lin and W.
Strauss. The novelty of this work is in
the extent of the truncation: we transform the problem into a
finite-dimensional problem which is rather easy to solve. We then
retrieve the original problem by letting the truncation parameter
ntend to ∞.
Advisors/Committee Members: Strauss, Walter (Director), Guo, Yan (Reader), Holmer, Justin (Reader).
Subjects/Keywords: kinetic theory
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APA (6th Edition):
Ben-Artzi, J. (2011). Linear Instability of Nonmonotone Super Heated
Plasmas. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:11412/
Chicago Manual of Style (16th Edition):
Ben-Artzi, Jonathan. “Linear Instability of Nonmonotone Super Heated
Plasmas.” 2011. Doctoral Dissertation, Brown University. Accessed April 15, 2021.
https://repository.library.brown.edu/studio/item/bdr:11412/.
MLA Handbook (7th Edition):
Ben-Artzi, Jonathan. “Linear Instability of Nonmonotone Super Heated
Plasmas.” 2011. Web. 15 Apr 2021.
Vancouver:
Ben-Artzi J. Linear Instability of Nonmonotone Super Heated
Plasmas. [Internet] [Doctoral dissertation]. Brown University; 2011. [cited 2021 Apr 15].
Available from: https://repository.library.brown.edu/studio/item/bdr:11412/.
Council of Science Editors:
Ben-Artzi J. Linear Instability of Nonmonotone Super Heated
Plasmas. [Doctoral Dissertation]. Brown University; 2011. Available from: https://repository.library.brown.edu/studio/item/bdr:11412/
6.
Malik, Numann.
Dark soliton linearization of the 1D Gross-Pitaevskii
equation.
Degree: Department of Mathematics, 2018, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:792705/
► We study the one-dimensional Gross-Pitaevskii equation, a cubic defocusing non-linear Schrodinger equation with nonvanishing boundary conditions. In particular we linearize around the dark solitons, which…
(more)
▼ We study the one-dimensional Gross-Pitaevskii
equation, a cubic defocusing non-linear Schrodinger equation with
nonvanishing boundary conditions. In particular we linearize around
the dark solitons, which are a family of exact solutions that do
not decay at spatial infinity (as opposed to bright solitons in the
focusing NLS). The dark solitons we study are exact solutions of
the Gross-Pitaevskii equation, which has been shown to be
completely integrable by means of the inverse scattering transform.
In particular we linearize around these solitons to produce
specific matrix operators which contain important spectral data.
Such information is understood by discovering the main ingredients
to build up distorted Fourier transforms and projections on to
eigenvalues. These are the Jost functions, namely bounded solutions
to the eigenvalue problem, which are then used to compute the
resolvent kernel. We give a comprehensive description of the
long-time dynamics exhibited by perturbations of the black soliton
after looking carefully at the vacuum steady state case (the
‘whitest’ dark soliton 1). This is very informative as the constant
coefficient problem has many similarities to the more generic black
soliton case. In particular they both give rise to singular
behavior at zero energy. So when studying the evolution of the
perturbation we observe special asymptotics at low frequencies.
Finally we derive the scattering theory for the matrix differential
operator in the more general gray soliton case. This is motivated
by the fact that understanding certain properties of this operator
can lead to a new proof for orbital stability.
Advisors/Committee Members: Holmer, Justin (Advisor), Strauss, Walter (Reader), Pausader, Benoit (Reader).
Subjects/Keywords: Differential equations; Partial
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APA (6th Edition):
Malik, N. (2018). Dark soliton linearization of the 1D Gross-Pitaevskii
equation. (Thesis). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:792705/
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Malik, Numann. “Dark soliton linearization of the 1D Gross-Pitaevskii
equation.” 2018. Thesis, Brown University. Accessed April 15, 2021.
https://repository.library.brown.edu/studio/item/bdr:792705/.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Malik, Numann. “Dark soliton linearization of the 1D Gross-Pitaevskii
equation.” 2018. Web. 15 Apr 2021.
Vancouver:
Malik N. Dark soliton linearization of the 1D Gross-Pitaevskii
equation. [Internet] [Thesis]. Brown University; 2018. [cited 2021 Apr 15].
Available from: https://repository.library.brown.edu/studio/item/bdr:792705/.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Malik N. Dark soliton linearization of the 1D Gross-Pitaevskii
equation. [Thesis]. Brown University; 2018. Available from: https://repository.library.brown.edu/studio/item/bdr:792705/
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
7.
Hong, Younghun.
Nonlinear Schrödinger Equations with Potentials.
Degree: PhD, Mathematics, 2013, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:320622/
► In this work, we develop new harmonic analytic tools to study local and global well-posedness of nonlinear Schrödinger equations perturbed by a general class of…
(more)
▼ In this work, we develop new harmonic analytic tools
to study local and global well-posedness of nonlinear Schrödinger
equations perturbed by a general class of short range of
potentials. First, we prove a spectral multiplier theorem which
holds even for Schrödinger operators having negative eigenvalues.
As an application, we obtain local well-posedness in the energy
space. Second, we prove global well-posedness below the energy
norms for a 3d cubic defocusing nonlinear Schrödinger equation.
Employing the Littlewood-Paley projections associated with a
Schrödinger operator and the structure formula for the wave
operator, the program of Colliander-Keel-Staffilani-Takaoka-Tao,
called the I-method, is run in the perturbed setting.
Advisors/Committee Members: Holmer, Justin (Director), Strauss, Walter (Reader), Chen, Xuwen (Reader).
Subjects/Keywords: Nonlinear Schrödinger Equation
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to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Hong, Y. (2013). Nonlinear Schrödinger Equations with Potentials. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:320622/
Chicago Manual of Style (16th Edition):
Hong, Younghun. “Nonlinear Schrödinger Equations with Potentials.” 2013. Doctoral Dissertation, Brown University. Accessed April 15, 2021.
https://repository.library.brown.edu/studio/item/bdr:320622/.
MLA Handbook (7th Edition):
Hong, Younghun. “Nonlinear Schrödinger Equations with Potentials.” 2013. Web. 15 Apr 2021.
Vancouver:
Hong Y. Nonlinear Schrödinger Equations with Potentials. [Internet] [Doctoral dissertation]. Brown University; 2013. [cited 2021 Apr 15].
Available from: https://repository.library.brown.edu/studio/item/bdr:320622/.
Council of Science Editors:
Hong Y. Nonlinear Schrödinger Equations with Potentials. [Doctoral Dissertation]. Brown University; 2013. Available from: https://repository.library.brown.edu/studio/item/bdr:320622/
8.
Zhang, Xiangxiong.
Maximum-Principle-Satisfying and Positivity-Preserving High
Order Schemes for Conservation Laws.
Degree: PhD, Mathematics, 2011, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:11274/
► This dissertation presents high order schemes for conservation laws which are, total-variation-diminishing for the one-dimensional scalar case,maximum-principle-satisfying for the multi-dimensional scalar case and positivity-preserving for…
(more)
▼ This dissertation presents high order schemes for
conservation laws which are, total-variation-diminishing for the
one-dimensional scalar case,maximum-principle-satisfying for the
multi-dimensional scalar case and positivity-preserving for systems
such as compressible Euler equations. For numerical schemes solving
scalar conservation laws, the maximum principle is a desired
property because numerical solutions violating it might be
physically meaningless. Such schemes for multi-dimensional problems
in the literature were at most second order accurate. Exact time
evolution can be used to construct maximum-principle-satisfying
high order schemes for the one-dimensional case but generalizations
to multi-dimension are very difficult. We propose a new general
framework to construct maximum-principle-satisfying high order
discontinuous Galerkin method and finite volume schemes by using
strong stability preserving (SSP) Runge-Kutta or multi-step time
discretization. The main advantage of this method includes
straightforward multi-dimensional generalizations on both
structured and unstructured meshes. This is the first time that
genuinely high order schemes are obtained satisfying a strict
maximum principle especially for multi-dimensional nonlinear scalar
conservation laws.The same idea can be used to construct high order
schemes preserving the positivity of certain physical quantities,
such as density and pressure for compressible Euler equations,
water height for shallow water equations, and density for Vlasov
transport equations. These schemes have been applied in
computational fluid dynamics, computational astronomy and
astrophysics, plasma simulation, and traffic flow
models.
Advisors/Committee Members: Shu, Chi-Wang (Director), Strauss, Walter (Reader), Hesthaven, Jan (Reader).
Subjects/Keywords: Maximum-Principle-Satisfying
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APA ·
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MLA ·
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Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Zhang, X. (2011). Maximum-Principle-Satisfying and Positivity-Preserving High
Order Schemes for Conservation Laws. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:11274/
Chicago Manual of Style (16th Edition):
Zhang, Xiangxiong. “Maximum-Principle-Satisfying and Positivity-Preserving High
Order Schemes for Conservation Laws.” 2011. Doctoral Dissertation, Brown University. Accessed April 15, 2021.
https://repository.library.brown.edu/studio/item/bdr:11274/.
MLA Handbook (7th Edition):
Zhang, Xiangxiong. “Maximum-Principle-Satisfying and Positivity-Preserving High
Order Schemes for Conservation Laws.” 2011. Web. 15 Apr 2021.
Vancouver:
Zhang X. Maximum-Principle-Satisfying and Positivity-Preserving High
Order Schemes for Conservation Laws. [Internet] [Doctoral dissertation]. Brown University; 2011. [cited 2021 Apr 15].
Available from: https://repository.library.brown.edu/studio/item/bdr:11274/.
Council of Science Editors:
Zhang X. Maximum-Principle-Satisfying and Positivity-Preserving High
Order Schemes for Conservation Laws. [Doctoral Dissertation]. Brown University; 2011. Available from: https://repository.library.brown.edu/studio/item/bdr:11274/
9.
Walsh, Samuel Peter.
Stratified and steady periodic water waves.
Degree: PhD, Applied Mathematics, 2010, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:11084/
► This thesis considers two-dimensional stratified water waves propagating under the force of gravity over an impermeable flat bed and with a free surface. In the…
(more)
▼ This thesis considers two-dimensional stratified water
waves propagating under the force of gravity over an impermeable
flat bed and with a free surface. In the absence of surface
tension, it is proved that there exists of a global continuum of
classical solutions that are periodic and traveling. These waves,
moreover, can exhibit large density variation, speed, and
amplitude. When the motion is assumed to be driven by capillarity
on the surface and a gravitational force acting on the body of the
fluid, it is shown that there exists global continua of such
solutions. In both regimes, this is accomplished by first
constructing a 1-parameter family of laminar flow solutions, then
applying bifurcation theory methods to obtain local curves of small
amplitude solutions branching from the laminar curve at an
eigenvalue of the linearized problem. Each solution curve is then
continued globally by means of a degree theoretic argument in the
spirit of Rabinowitz. We also provide an alternate global
bifurcation theorem via the analytic continuation method of
Dancer.Finally, we consider the question of symmetry for
two-dimensional stably stratified steady periodic gravity water
waves with surface profiles monotonic between crests and troughs.
We provide sufficient conditions under which such waves are
necessarily symmetric. We do this by first exploiting some elliptic
structure in the governing equations to show that, in certain size
regimes, a maximum principle holds. This then forms the basis for a
method of moving planes argument.
Advisors/Committee Members: Strauss, Walter (Director), Dafermos, Constantine (Reader), Mallet-Paret, John (Reader).
Subjects/Keywords: partial differential equations
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Manager
APA (6th Edition):
Walsh, S. P. (2010). Stratified and steady periodic water waves. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:11084/
Chicago Manual of Style (16th Edition):
Walsh, Samuel Peter. “Stratified and steady periodic water waves.” 2010. Doctoral Dissertation, Brown University. Accessed April 15, 2021.
https://repository.library.brown.edu/studio/item/bdr:11084/.
MLA Handbook (7th Edition):
Walsh, Samuel Peter. “Stratified and steady periodic water waves.” 2010. Web. 15 Apr 2021.
Vancouver:
Walsh SP. Stratified and steady periodic water waves. [Internet] [Doctoral dissertation]. Brown University; 2010. [cited 2021 Apr 15].
Available from: https://repository.library.brown.edu/studio/item/bdr:11084/.
Council of Science Editors:
Walsh SP. Stratified and steady periodic water waves. [Doctoral Dissertation]. Brown University; 2010. Available from: https://repository.library.brown.edu/studio/item/bdr:11084/
10.
Lin, Quanhui.
On the interactions of dispersive modes and soliton
dynamics.
Degree: PhD, Mathematics, 2012, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:297679/
► A soliton generally refers to a particle-like solution, i.e. a localized solution of finite energy, that does not change its shape in propagation. It is…
(more)
▼ A soliton generally refers to a particle-like
solution, i.e. a localized solution of finite energy, that does not
change its shape in propagation. It is caused by a cancelation of
nonlinear and dispersive effects in the medium. This thesis studies
the behaviors of solitons under several types of environments. In
Chapter 1, we show that, for the 1d cubic NLS equation, widely
separated equal amplitude in-phase solitons attract and
opposite-phase solitons repel. The result gives an exact
description of the evolution of the two solitons valid until the
solitons have moved a distance comparable to the logarithm of the
initial separation. We present as a special case of a general
framework which also addresses, for example, the dynamics of single
solitons subject to external forces. In Chapter 2, we consider the
interaction of a NLS soliton with an attractive delta potential,
and show that slow incoming solitons will be reflected by the
potential well. This gives a partial justification of the quantum
reflection phenomenon studied numerically or experimentally in many
physics papers. We mainly follow the framework of Chapter 1. In
Chapter 3, we consider the dynamics of solitons to the perturbed
mKdV equation \partial
t u = \partial
x(-\partial
x2 u - 2u
3) +
ε Vu, where V∈ C
1b and ε is small. This type
of perturbation is non-Hamiltonian, and tends to give rise to
significant dispersive radiation, which affects the eficiency of
applying the framework in Chapter 1. We here adapt the Martel-Merle
local virial estimate to supplement the energy Lyapunov estimate.
This approach was initially employed by Holmer to study the KdV
equation with a slowly varying potential.
Advisors/Committee Members: Holmer, Justin (Director), Holmer, Justin (Reader), Strauss, Walter (Reader), Guo, Yan (Reader).
Subjects/Keywords: Dispersive Equations
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APA (6th Edition):
Lin, Q. (2012). On the interactions of dispersive modes and soliton
dynamics. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:297679/
Chicago Manual of Style (16th Edition):
Lin, Quanhui. “On the interactions of dispersive modes and soliton
dynamics.” 2012. Doctoral Dissertation, Brown University. Accessed April 15, 2021.
https://repository.library.brown.edu/studio/item/bdr:297679/.
MLA Handbook (7th Edition):
Lin, Quanhui. “On the interactions of dispersive modes and soliton
dynamics.” 2012. Web. 15 Apr 2021.
Vancouver:
Lin Q. On the interactions of dispersive modes and soliton
dynamics. [Internet] [Doctoral dissertation]. Brown University; 2012. [cited 2021 Apr 15].
Available from: https://repository.library.brown.edu/studio/item/bdr:297679/.
Council of Science Editors:
Lin Q. On the interactions of dispersive modes and soliton
dynamics. [Doctoral Dissertation]. Brown University; 2012. Available from: https://repository.library.brown.edu/studio/item/bdr:297679/
. 
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