146 research outputs found
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Time-periodic solutions of the Benjamin-Ono equation
We present a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which, in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase and the real part of one of the Fourier modes at t = 0. We use our method to study global paths of non-trivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached (or until the solution blows up). By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODE's governing the evolution of solitons using the ansatz suggested by the numerical simulations
Large time wellposdness to the 3-D Capillary-Gravity Waves in the long wave regime
In the regime of weakly transverse long waves, given long-wave initial data,
we prove that the nondimensionalized water wave system in an infinite strip
under influence of gravity and surface tension on the upper free interface has
a unique solution on [0,{T}/\eps] for some \eps independent of constant
We shall prove in the subsequent paper \cite{MZZ2} that on the same time
interval, these solutions can be accurately approximated by sums of solutions
of two decoupled Kadomtsev-Petviashvili (KP) equations.Comment: Split the original paper(The long wave approximation to the 3-D
capillary-gravity waves) into two parts, this is the first on
Well-posedness of Hydrodynamics on the Moving Elastic Surface
The dynamics of a membrane is a coupled system comprising a moving elastic
surface and an incompressible membrane fluid. We will consider a reduced
elastic surface model, which involves the evolution equations of the moving
surface, the dynamic equations of the two-dimensional fluid, and the
incompressible equation, all of which operate within a curved geometry. In this
paper, we prove the local existence and uniqueness of the solution to the
reduced elastic surface model by reformulating the model into a new system in
the isothermal coordinates. One major difficulty is that of constructing an
appropriate iterative scheme such that the limit system is consistent with the
original system.Comment: The introduction is rewritte
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Track A Basic Science
Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/138319/1/jia218438.pd
The evolution of lung cancer and impact of subclonal selection in TRACERx
Lung cancer is the leading cause of cancer-associated mortality worldwide1. Here we analysed 1,644 tumour regions sampled at surgery or during follow-up from the first 421 patients with non-small cell lung cancer prospectively enrolled into the TRACERx study. This project aims to decipher lung cancer evolution and address the primary study endpoint: determining the relationship between intratumour heterogeneity and clinical outcome. In lung adenocarcinoma, mutations in 22 out of 40 common cancer genes were under significant subclonal selection, including classical tumour initiators such as TP53 and KRAS. We defined evolutionary dependencies between drivers, mutational processes and whole genome doubling (WGD) events. Despite patients having a history of smoking, 8% of lung adenocarcinomas lacked evidence of tobacco-induced mutagenesis. These tumours also had similar detection rates for EGFR mutations and for RET, ROS1, ALK and MET oncogenic isoforms compared with tumours in never-smokers, which suggests that they have a similar aetiology and pathogenesis. Large subclonal expansions were associated with positive subclonal selection. Patients with tumours harbouring recent subclonal expansions, on the terminus of a phylogenetic branch, had significantly shorter disease-free survival. Subclonal WGD was detected in 19% of tumours, and 10% of tumours harboured multiple subclonal WGDs in parallel. Subclonal, but not truncal, WGD was associated with shorter disease-free survival. Copy number heterogeneity was associated with extrathoracic relapse within 1âyear after surgery. These data demonstrate the importance of clonal expansion, WGD and copy number instability in determining the timing and patterns of relapse in non-small cell lung cancer and provide a comprehensive clinical cancer evolutionary data resource
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