519 research outputs found
Separation of variables in quasi-potential systems of bi-cofactor form
We perform variable separation in the quasi-potential systems of equations of
the form {}, where
and are Killing tensors, by embedding these systems into a
bi-Hamiltonian chain and by calculating the corresponding Darboux-Nijenhuis
coordinates on the symplectic leaves of one of the Hamiltonian structures of
the system. We also present examples of the corresponding separation
coordinates in two and three dimensions.Comment: LaTex, 30 pages, to appear in J. Phys. A: Math. Ge
Formation and Dynamics of Shock Waves in the Degasperis-Procesi Equation
Solutions of the DegasperisâProcesi nonlinear wave equation may develop discontinuities in finite time. As shown by Coclite and Karlsen, there is a uniquely determined entropy weak solution which provides a natural continuation of the solution past such a point. Here we study this phenomenon in detail for solutions involving interacting peakons and antipeakons. We show that a jump discontinuity forms when a peakon collides with an antipeakon, and that the entropy weak solution in this case is described by a "shockpeakon" ansatz reducing the PDE to a system of ODEs for positions, momenta, and shock strengths
Continuous and discontinuous piecewise linear solutions of the linearly forced inviscid Burgers equation
We study a class of piecewise linear solutions to the inviscid Burgers
equation driven by a linear forcing term. Inspired by the analogy with peakons,
we think of these solutions as being made up of solitons situated at the
breakpoints. We derive and solve ODEs governing the soliton dynamics, first for
continuous solutions, and then for more general shock wave solutions with
discontinuities. We show that triple collisions of solitons cannot take place
for continuous solutions, but give an example of a triple collision in the
presence of a shock.Comment: To appear in Journal of Nonlinear Mathematical Physics (proceedings
of NEEDS 2007). 16 pages, 3 figures, LaTeX + AMS packages + pstrick
The inverse spectral problem for the discrete cubic string
Given a measure on the real line or a finite interval, the "cubic string"
is the third order ODE where is a spectral parameter. If
equipped with Dirichlet-like boundary conditions this is a nonselfadjoint
boundary value problem which has recently been shown to have a connection to
the Degasperis-Procesi nonlinear water wave equation. In this paper we study
the spectral and inverse spectral problem for the case of Neumann-like boundary
conditions which appear in a high-frequency limit of the Degasperis--Procesi
equation. We solve the spectral and inverse spectral problem for the case of
being a finite positive discrete measure. In particular, explicit
determinantal formulas for the measure are given. These formulas generalize
Stieltjes' formulas used by Krein in his study of the corresponding second
order ODE .Comment: 24 pages. LaTeX + iopart, xypic, amsthm. To appear in Inverse
Problems (http://www.iop.org/EJ/journal/IP
The Cauchy two-matrix model
We introduce a new class of two(multi)-matrix models of positive Hermitean
matrices coupled in a chain; the coupling is related to the Cauchy kernel and
differs from the exponential coupling more commonly used in similar models. The
correlation functions are expressed entirely in terms of certain biorthogonal
polynomials and solutions of appropriate Riemann-Hilbert problems, thus paving
the way to a steepest descent analysis and universality results. The
interpretation of the formal expansion of the partition function in terms of
multicolored ribbon-graphs is provided and a connection to the O(1) model. A
steepest descent analysis of the partition function reveals that the model is
related to a trigonal curve (three-sheeted covering of the plane) much in the
same way as the Hermitean matrix model is related to a hyperelliptic curve.Comment: 34 pages, 2 figures. V2: changes only to metadat
The Degasperis-Procesi equation with self-consistent sources
The Degasperis-Procesi equation with self-consistent sources(DPESCS) is
derived. The Lax representation and the conservation laws for DPESCS are
constructed. The peakon solution of DPESCS is obtained.Comment: 15 page
Multi loop soliton solutions and their interactions in the Degasperis-Procesi equation
In this article, we construct loop soliton solutions and mixed soliton - loop
soliton solution for the Degasperis-Procesi equation. To explore these
solutions we adopt the procedure given by Matsuno. By appropriately modifying
the -function given in the above paper we derive these solutions. We
present the explicit form of one and two loop soliton solutions and mixed
soliton - loop soliton solutions and investigate the interaction between (i)
two loop soliton solutions in different parametric regimes and (ii) a loop
soliton with a conventional soliton in detail.Comment: Published in Physica Scripta (2012
Allelic expression mapping across cellular lineages to establish impact of non-coding SNPs
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