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Register a new userPeakons and Cauchy Biorthogonal Polynomials
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The contents of the paper is now covered in two separate papers arXiv:0904.2188 and arXiv:0904.2602. Please refer to those. Note that you can still access the original version arXiv:0711.4082v1.
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We consider the Cauchy problem for the Burgers hierarchy with general time dependent coefficients. The closed form for the Greens function of the corresponding linear equation of arbitrary order $N$ is shown to be a sum of generalised hypergeometric functions. For suitably damped initial conditions we plot the time dependence of the Cauchy problem over a range of $N$ values. For $N=1$, we introduce a spatial forcing term. Using connections between the associated second order linear Schr{o}dinger and Fokker-Planck equations, we give closed form expressions for the corresponding Greens functions of the sinked Bessel process with constant drift. We then apply the Greens function to give time dependent profiles for the corresponding forced Burgers Cauchy problem.
The Riemann-Hilbert problems for multiple orthogonal polynomials of types I and II are used to derive string equations associated to pairs of Lax-Orlov operators. A method for determining the quasiclassical limit of string equations in the phase space of the Whitham hierarchy of dispersionless integrable systems is provided. Applications to the analysis of the large-n limit of multiple orthogonal polynomials and their associated random matrix ensembles and models of non-intersecting Brownian motions are given.
We introduce two classes of discrete polynomials and construct discrete equations admitting a Lax representation in terms of these polynomials. Also we give an approach which allows to construct lattice integrable hierarchies in its explicit form and show some examples.
In this paper we investigate the Q-ball Ansatz in the baby Skyrme model. First, the appearance of peakons, i.e. solutions with extremely large absolute values of the second derivative at maxima, is analyzed. It is argued that such solutions are intrinsic to the baby Skyrme model and do not depend on the detailed form of a potential used in calculations. Next, we concentrate on compact non spinning Q-balls. We show the failure of a small parameter expansion in this case. Finally, we explore the existence and parameter dependence of Q-ball solutions.
The Degasperis-Procesi equation can be derived as a member of a one-parameter family of asymptotic shallow water approximations to the Euler equations with the same asymptotic accuracy as that of the Camassa-Holm equation. In this paper, we study the orbital stability problem of the peaked solitons to the Degasperis-Procesi equation on the line. By constructing a Liapunov function, we prove that the shapes of these peakon solitons are stable under small perturbations.
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