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Register a new userIsospectral flows for the inhomogeneous string density problem
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We derive isospectral flows of the mass density in the string boundary value problem corresponding to general boundary conditions. In particular, we show that certain class of rational flows produces in a suitable limit all flows generated by polynomials in negative powers of the spectral parameter. We illustrate the theory with concrete examples of isospectral flows of discrete mass densities which we prove to be Hamiltonian and for which we provide explicit solutions of equations of motion in terms of Stieltjes continued fractions and Hankel determinants.
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We establish the Lagrangian nature of the discrete isospectral and isomonodromic dynamical systems corresponding to the re-factorization transformations of the rational matrix functions on the Riemann sphere. Specifically, in the isospectral case we generalize the Moser-Veselov approach to integrability of discrete systems via the re-factorization of matrix polynomials to a more general class of matrix rational functions that have a simple divisor and, in the quadratic case, explicitly write the Lagrangian function for such systems. Next we show that if we let certain parameters in this Lagrangian to be time-dependent, the resulting Euler-Lagrange equations describe the isomonodromic transformations of systems of linear difference equations. It is known that in some special cases such equations reduce to the difference Painleve equation. As an example, we show how to obtain the difference Painlev`e V equation in this way, and hence we establish that this equation can be written in the Lagrangian form.
We provide a list of explicit eigenfunctions of the trigonometric Calogero-Sutherland Hamiltonian associated to the root system of the exceptional Lie algebra E8. The quantum numbers of these solutions correspond to the first and second order weights of the Lie algebra.
For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the Ernst equation. By solving this equation with a {it Dirichlet} boundary condition imposed along the disk, Neugebauer and Meinel in the 1990s famously derived an explicit expression for the spacetime metric corresponding to the Bardeen-Wagoner uniformly rotating disk of dust. In this paper, we consider a similar boundary value problem for a rotating disk in which a {it Neumann} boundary condition is imposed along the disk instead of a Dirichlet condition. Using the integrable structure of the Ernst equation, we are able to reduce the problem to a Riemann-Hilbert problem on a genus one Riemann surface. By solving this Riemann-Hilbert problem in terms of theta functions, we obtain an explicit expression for the Ernst potential. Finally, a Riemann surface degeneration argument leads to an expression for the associated spacetime metric.
A trajectory isomorphism between the two Newtonian fixed center problem in the sphere and two associated planar two fixed center problems is constructed by performing two simultaneous gnomonic projections in $S^2$. This isomorphism converts the original quadratures into elliptic integrals and allows the bifurcation diagram of the spherical problem to be analyzed in terms of the corresponding ones of the planar systems. The dynamics along the orbits in the different regimes for the problem in $S^2$ is expressed in terms of Jacobi elliptic functions.
The focusing Nonlinear Schrodinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of NLS (what we call the Cauchy problem of the AWs), in the case of a finite number $N$ of unstable modes. We show how the finite gap method adapts to this specific Cauchy problem through three basic simplifications, allowing one to construct the solution, at the leading and relevant order, in terms of elementary functions of the initial data. More precisely, we show that, at the leading order, i) the initial data generate a partition of the time axis into a sequence of finite intervals, ii) in each interval $I$ of the partition, only a subset of ${cal N}(I)le N$ unstable modes are visible, and iii) the NLS solution is approximated, for $tin I$, by the ${cal N}(I)$-soliton solution of Akhmediev type, describing the nonlinear interaction of these visible unstable modes, whose parameters are expressed in terms of the initial data through elementary functions. This result explains the relevance of the $m$-soliton solutions of Akhmediev type, with $mle N$, in the generic periodic Cauchy problem of the AWs, in the case of a finite number $N$ of unstable modes.
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