Publications

Hopf-Like Bifurcation in a Wave Equation at a Removable Singularity

Published in Submitted, 2022

It is shown that a one-dimensional damped wave equation with an odd time derivative nonlinearity exhibits small amplitude bifurcating time periodic solutions, when the bifurcation parameter is the linear damping coefficient is positive and accumulates to zero. The upshot is that the singularity of the linearized operator at criticality which stems from the well known small divisor problem for the wave operator, is entirely removed without the need to exclude parameters via Diophantine conditions, nor the use of accelerated convergence schemes. Only the contraction mapping principle is used.

Recommended citation: N. Kosovalic and B. Pigott, "Hopf-Like Bifurcation in a Wave Equation at a Removable Singularity." Submitted. (2022).'

Symmetric vibrations of higher dimensional nonlinear wave equations

Published in Selecta Mathematica (N.S.), 2022

We prove a result characterizing conditions for the existence and uniqueness of solutions of a certain Diophantine equation, then using techniques from equivariant bifurcation theory, we apply the result to prove symmetric Hopf bifurcation type theorems for both dissipative and non-dissipative autonomous wave equations, for a large set of spatial dimensions. For the latter only the classical implicit function theorem is used. The set of admissible spatial dimensions is the union of the perfect squares together with finitely many non-perfect squares.

Recommended citation: N. Kosovalic and B. Pigott, "Symmetric vibrations of higher dimensional nonlinear wave equations." Selecta Math. (N.S.). 28 (3) (2022). https://doi.org/10.1007/s00029-022-00761-7

Uniqueness for sums of nonvanishing squares

Published in Integers, 2020

We address the issue of uniqueness for sums of nonvanishing squares; that is, we determine all positive integers N that can be represented as a sum of k ≥ 5 nonvanishing squares in essentially only one way. Our methods are elementary and are based on a lower bound on the number of 1s that must be present in such a representation.

Recommended citation: N. Kosovalic and B. Pigott, "Uniqueness for sums of nonvanishing squares." Integers. 20 (2020). http://math.colgate.edu/~integers/u93/u93.pdf

Self-excited vibrations for damped and delayed higher dimensional wave equations

Published in Discrete and Continuous Dynamical Systems. Series A, 2019

In the article [12] it is shown that time delay induces self-excited vibrations in a one dimensional damped wave equation. Here we generalize this result for higher spatial dimensions. We prove the existence of branches of nontrivial time periodic solutions for spatial dimensions $ d\ge 2 $. For $ d> 2 $, the bifurcating periodic solutions have a fixed spatial frequency vector, which is the solution of a certain Diophantine equation. The case $ d = 2 $ must be treated separately from the others. In particular, it is shown that an arbitrary number of symmetry breaking orbitally distinct time periodic solutions exist, provided $ d $ is big enough, with respect to the symmetric group action. The direction of bifurcation is also obtained.

Recommended citation: N. Kosovalic and B. Pigott, "Self-excited vibrations for damped and delayed higher dimensional wave equations." Discrete Contin. Dyn. Syst.. 39 (5) (2019). http://dx.doi.org/10.3934/dcds.2019102

Self-excited vibrations for damped and delayed 1-dimensional wave equations

Published in Journal of Dynamics and Differential Equations, 2019

It is shown that time delay induces self-excited vibrations in one dimensional damped wave equations, via Hopf bifurcation. In order to obtain classical solutions and $𝐶^{\infty}$ smoothness with respect to the amplitude parameter, we avoid the use of any abstract Hopf bifurcation theorem, and instead work directly in Sobolev spaces on the 2-torus. As a result, a “derivative loss” problem arises, which is overcome using the regularizing effects of the time delay and damping. Only the classical implicit function theorem is used. The direction of bifurcation is also obtained.

Recommended citation: N. Kosovalic and B. Pigott, "Self-excited vibrations for damped and delayed 1-dimensional wave equations." J. Dynam. Differential Equations. 31 (2019). https://doi.org/10.1007/s10884-018-9654-2

Nonlinear profile decomposition and the concentration phenomenon for supercritical generalized KdV equations

Published in Indiana University Mathematics Journal, 2018

A nonlinear profile decomposition is established for solutions of supercritical generalized Korteweg-de Vries equations. As a consequence, we obtain a concentration result for finite-time blow-up solutions that are of type II.

Recommended citation: L.G. Farah and B. Pigott, "Nonlinear profile decomposition and the concentration phenomenon for supercritical generalized KdV equations." Indiana Univ. Math. J.. 67 (5) (2018). https://doi.org/10.1512/iumj.2018.67.7471

Long-term stability for KdV solitons in weighted $H^{s}$ spaces

Published in Communications on Pure and Applied Analysis, 2017

In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the I-method and spectral analysis following Pego and Weinstein, we are able to show that, in the exponentially weighted space, the perturbation of a soliton decays exponentially for arbitrarily long times. The finite time restriction is due to a lack of global control of the unweighted perturbation.

Recommended citation: B. Pigott and S. Raynor, "Long-term stability for KdV solitons in weighted $H^{s}$ spaces." Commun. Pure Appl. Anal.. 16 (2) (2017). https://doi.org/10.3934/cpaa.2017020

On mass concentration for the critical generalized Korteweg-de Vries equation

Published in Proceedings of the Edinburgh Mathematical Society. Series II, 2016

We show that blow-up solutions of the critical generalized Korteweg–de Vries equation in $H^{1}(\mathbb{R})$ concentrate at least the mass of the ground state at the blow-up time. The I-method is used to prove a slightly weaker result in $H^{s}(\mathbb{R})$ with $16/17 < s < 1$. Under an assumption on the precise blow-up rate, we are able to use similar arguments to prove a more precise analogue of the $H^{1}(\mathbb{R})$ concentration result over the same range of $s$.

Recommended citation: B. Pigott, "On mass concentration for the critical generalized Korteweg-de Vries equation." Proc. Edinb. Math. Soc. (2). 59 (2) (2016). https://doi.org/10.1017/S001309151500019X

Asymptotic stability for KdV solitons on weighted spaces via iteration

Published in Illinois Journal of Mathematics, 2014

In this paper, we reconsider the well-known result of Pego–Weinstein ( Comm. Math. Phys. 2 (1994) 305–349) that soliton solutions to the Korteweg–de Vries equation are asymptotically stable in exponentially weighted spaces. In this work, we recreate this result in the setting of modern well-posedness function spaces. We obtain asymptotic stability in the exponentially weighted space via an iteration argument. Our purpose here is to lay the groundwork to use the I-method to obtain asymptotic stability below $H^{1}$, which will be done in a second, forthcoming paper (Asymptotic stability for KdV solitons in weighted H^{s} spaces). This will be possible because the exponential approach rate obtained here will defeat the polynomial loss in traditional applications of the $I$-method ( Commun. Pure Appl. Anal. 2 (2003) 277–296, Discrete Contin. Dyn. Syst. 9 (2003) 31–54, Commun. Pure Appl. Anal. 13 (2014) 389–418).

Recommended citation: B. Pigott and S. Raynor, "Asymptotic stability for KdV solitons in weighted spaces via iteration." Illinois J. Math.. 58 (2) (2014). http://projecteuclid.org/euclid.ijm/1436275493

Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations

Published in Communications on Pure and Applied Analysis, 2014

We prove polynomial-in-time upper bounds for the orbital instability of solitons for subcritical generalized Korteweg-de Vries equations in $H_{x}^{s}(\mathbb{R})$ with $s < 1$. By combining coercivity estimates of Weinstein with the $I$-method as developed by Colliander, Keel, Staffilani, Takaoka, and Tao, we construct a modified energy functional which is shown to be almost conserved while providing us with an estimate of the deviation of the solution from the ground state curve. The iteration of the almost conservation law for the modified energy functional over time intervals of uniform length yields the polynomial upper bound.

Recommended citation: B. Pigott, "Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations." Commun. Pure Appl. Anal.. 13 (1) (2014). https://doi.org/10.3934/cpaa.2014.13.389

Low Regularity Stability for Subcritical Generalized Korteweg-de Vries Equations

Published in Ph.D. Thesis - University of Toronto, 2011

In this thesis we prove polynomial-in-time upper bounds for the orbital instability of solitons for subcritical generalized Korteweg-de Vries equations in $H^{s}_{x}(\mathbb{R})$ with $s < 1$. By combining coercivity estimates of Weinstein with the $I$-method as developed by Colliander, Keel, Staffilani, Takaoka, and Tao, we construct a modified energy functional which is shown to be almost conserved while providing us with an estimate of the deviation of the solution from the ground state curve. The iteration of the almost conservation law for the modified energy functional over time intervals of uniform length yields the polynomial upper bound.

Recommended citation: B. Pigott, "Low Regularity Stability for Subcritical Generalized Korteweg-de Vries Equations." Ph.D. Thesis, University of Toronto. (2011). https://tspace.library.utoronto.ca/handle/1807/31899

The integral homology of orientable Seifert manifolds

Published in Topology and Its Applications, 2003

For any orientable Seifert manifold $M$, the integral homology group $H_{1}(M) = H_{1}(M;\mathbb{Z})$ is computed and explicit generators are found. This calculation gives a presentation for the $p$-torsion of $H_{1}(M)$ for any prime $p$. Since Seifert manifolds have dimension $3$, $H_{1}(M)$ determines $H_{\ast}(M;A)$ and $H^{\ast}(M;A)$ as well, for any abelian group $A$. The complete details are given when $A = \mathbb{Z},\mathbb{Z}/p^{s}$. In order to calculate the partition functions of the Dijkgraaf–Witten topological quantum field theories it is necessary to compute the linking form of the underlying $3$-manifold. In the case of the orientable Seifert manifolds it is possible to compute the linking form. The calculation of the linking form involves finding a presentation of the torsion of the first integral homology of the orientable Seifert manifolds, which is the main result of this paper.

Recommended citation: J. Bryden, T. Lawson, B. Pigott, and P. Zvengrowski, "The integral homology of orientable Seifert manifolds." Topology Appl.. 127 (1-2) (2003). https://doi.org/10.1016/S0166-8641(02)00062-7

Brian Pigott, PhD

Publications