33 research outputs found
A class of equations with peakon and pulson solutions (with an Appendix by Harry Braden and John Byatt-Smith)
We consider a family of integro-differential equations depending upon a
parameter as well as a symmetric integral kernel . When and
is the peakon kernel (i.e. up to rescaling) the
dispersionless Camassa-Holm equation results, while the Degasperis-Procesi
equation is obtained from the peakon kernel with . Although these two
cases are integrable, generically the corresponding integro-PDE is
non-integrable. However,for the family restricts to the pulson family of
Fringer & Holm, which is Hamiltonian and numerically displays elastic
scattering of pulses. On the other hand, for arbitrary it is still possible
to construct a nonlocal Hamiltonian structure provided that is the peakon
kernel or one of its degenerations: we present a proof of this fact using an
associated functional equation for the skew-symmetric antiderivative of .
The nonlocal bracket reduces to a non-canonical Poisson bracket for the peakon
dynamical system, for any value of .Comment: Contribution to volume of Journal of Nonlinear Mathematical Physics
in honour of Francesco Caloger
Heron triangles with two rational medians and Somos-5 sequences
Triangles with integer length sides and integer area are known as Heron
triangles. Taking rescaling freedom into account, one can apply the same name
when all sides and the area are rational numbers. A perfect triangle is a Heron
triangle with all three medians being rational, and it is a longstanding
conjecture that no such triangle exists. However, Buchholz and Rathbun showed
that there are infinitely many Heron triangles with two rational medians, an
infinite subset of which are associated with rational points on an elliptic
curve with Mordell-Weil group , and they observed a connection with a pair of Somos-5
sequences. Here we make the latter connection more precise by providing
explicit formulae for the integer side lengths, the two rational medians, and
the area in this infinite family of Heron triangles. The proof uses a combined
approach to Somos-5 sequences and associated Quispel-Roberts-Thompson (QRT)
maps in the plane, from several different viewpoints: complex analysis, real
dynamics, and reduction modulo a prime.Comment: Minor typos and one entry in Table 9 correcte
Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation
Recently Vladimir Novikov found a new integrable analogue of the Camassa-Holm
equation, admitting peaked soliton (peakon) solutions, which has nonlinear
terms that are cubic, rather than quadratic. In this paper, the explicit
formulas for multipeakon solutions of Novikov's cubically nonlinear equation
are calculated, using the matrix Lax pair found by Hone and Wang. By a
transformation of Liouville type, the associated spectral problem is related to
a cubic string equation, which is dual to the cubic string that was previously
found in the work of Lundmark and Szmigielski on the multipeakons of the
Degasperis-Procesi equation.Comment: 41 pages, LaTeX + AMS packages + pstrick
On the non-integrability of the Popowicz peakon system
We consider a coupled system of Hamiltonian partial differential equations
introduced by Popowicz, which has the appearance of a two-field coupling
between the Camassa-Holm and Degasperis-Procesi equations. The latter equations
are both known to be integrable, and admit peaked soliton (peakon) solutions
with discontinuous derivatives at the peaks. A combination of a reciprocal
transformation with Painlev\'e analysis provides strong evidence that the
Popowicz system is non-integrable. Nevertheless, we are able to construct exact
travelling wave solutions in terms of an elliptic integral, together with a
degenerate travelling wave corresponding to a single peakon. We also describe
the dynamics of N-peakon solutions, which is given in terms of an Hamiltonian
system on a phase space of dimension 3N.Comment: 8 pages, AIMS class file. Proceedings of AIMS conference on Dynamical
Systems, Differential Equations and Applications, Arlington, Texas, 200
Continued fractions for some transcendental numbers
We consider series of the form , where and the integer sequence satisfies a certain non-autonomous recurrence of second order, which entails that for n?1. It is shown that the terms of the sequence, and multiples of the ratios of successive terms, appear interlaced in the continued fraction expansion of the sum of the series, which is a transcendental number
Deformations of cluster mutations and invariant presymplectic forms
We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A_2: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A_3 and A_4 we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions of the discrete sine-Gordon equation
Two-component generalizations of the Camassa-Holm equation
A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators is carried out, which leads to bi-Hamiltonian structures for the same systems of equations. Some exact solutions and Lax pairs are also constructed for the systems considered
Discrete PainlevƩ equations from Y-systems
We consider T-systems and Y-systems arising from cluster mutations applied to quivers that have the property of being periodic under a sequence of mutations. The corresponding nonlinear recurrences for cluster variables (coefficient-free T-systems) were described in the work of Fordy and Marsh, who completely classified all such quivers in the case of period 1, and characterized them in terms of the skew-symmetric exchange matrix B that defines the quiver. A broader notion of periodicity in general cluster algebras was introduced by Nakanishi, who also described the corresponding Y-systems, and T-systems with coefficients.
A result of Fomin and Zelevinsky says that the coefficient-free T-system provides a solution of the Y-system. In this paper, we show that in general there is a discrepancy between these two systems, in the sense that the solution of the former does not correspond to the general solution of the latter. This discrepancy is removed by introducing additional non-autonomous coefficients into the T-system. In particular, we focus on the period 1 case and show that, when the exchange matrix B is degenerate, discrete Painlev\'e equations can arise from this construction
On the General Solution of the HeidemanāHogan Family of Recurrences
We consider a family of nonlinear rational recurrences of odd order which was introduced by Heideman and Hogan, and recently rediscovered in the theory of Laurent phenomenon algebras (a generalization of cluster algebras). All of these recurrences have the Laurent property, implying that for a particular choice of initial data (all initial values set to 1) they generate an integer sequence. For these particular sequences, Heideman and Hogan gave a direct proof of integrality by showing that the terms of the sequence also satisfy a linear recurrence relation with constant coefficients. Here we present an analogous result for the general solution of each of these recurrences