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A Chebyshev criterion with applications
https://ddd.uab.cat/record/228117
We show that a family of certain definite integrals forms a Chebyshev system if two families of associated functions appearing in their integrands are Chebyshev systems as well. We apply this criterion to several examples which appear in the context of perturbations of periodic non-autonomous ODEs to determine bounds on the number of isolated periodic solutions, as well as to persistence problems of periodic solutions for perturbed Hamiltonian systems. Gasull i Embid, ArmengolWed, 15 Jul 2020 12:00:36 GMThttps://ddd.uab.cat/record/2281172020Spectral stability of periodic waves in the generalized reduced Ostrovsky equation
https://ddd.uab.cat/record/221354
We consider stability of periodic travelling waves in the generalized reduced Ostrovsky equation with respect to co-periodic perturbations. Compared to the recent literature, we give a simple argument that proves spectral stability of all smooth periodic travelling waves independent of the nonlinearity power. The argument is based on the energy convexity and does not use coordinate transformations of the reduced Ostrovsky equations to the semi-linear equations of the Klein-Gordon type. Geyer, AnnaWed, 15 Apr 2020 15:23:09 GMThttps://ddd.uab.cat/record/2213542017On the number of limit cycles for perturbed pendulum equations
https://ddd.uab.cat/record/169482
We consider perturbed pendulum-like equations on the cylinder of the form x (x)= _=0^mQ_n, (x) x^ where Q_n, are trigonometric polynomials of degree n, and study the number of limit cycles that bifurcate from the periodic orbits of the unperturbed case =0 in terms of m and n. Our first result gives upper bounds on the number of zeros of its associated first order Melnikov function, in both the oscillatory and the rotary regions. These upper bounds are obtained expressing the corresponding Abelian integrals in terms of polynomials and the complete elliptic functions of first and second kind. Some further results give sharp bounds on the number of zeros of these integrals by identifying subfamilies which are shown to be Chebyshev systems. Gasull i Embid, ArmengolMon, 23 Jan 2017 15:21:47 GMThttps://ddd.uab.cat/record/1694822016Singular solutions for a class of traveling wave equations arising in hydrodynamics
https://ddd.uab.cat/record/169457
We give an exhaustive characterization of singular weak solutions for ordinary differential equations of the form 12^2 F'(u) =0, where F is an analytic function. Our motivation stems from the fact that in the context of hydrodynamics several prominent equations are reducible to an equation of this form upon passing to a moving frame. We construct peaked and cusped waves, fronts with finite-time decay and compact solitary waves. We prove that one cannot obtain peaked and compactly supported traveling waves for the same equation. In particular, a peaked traveling wave cannot have compact support and vice versa. To exemplify the approach we apply our results to the Camassa-Holm equation and the equation for surface waves of moderate amplitude, and show how the different types of singular solutions can be obtained varying the energy level of the corresponding planar Hamiltonian systems. Geyer, AnnaMon, 23 Jan 2017 15:21:46 GMThttps://ddd.uab.cat/record/1694572016Traveling surface waves of moderate amplitude in shallow water
https://ddd.uab.cat/record/150693
We study traveling wave solutions of an equation for surface waves of moderate amplitude arising as a shallow water approximation of the Euler equations for inviscid, incompressible and homogenous fluids. We obtain solitary waves of elevation and depression, including a family of solitary waves with compact support, where the amplitude may increase or decrease with respect to the wave speed. Our approach is based on techniques from dynamical systems and relies on a reformulation of the evolution equation as an autonomous Hamiltonian system which facilitates an explicit expression for bounded orbits in the phase plane to establish existence of the corresponding periodic and solitary traveling wave solutions. Gasull i Embid, ArmengolFri, 06 May 2016 08:59:50 GMThttps://ddd.uab.cat/record/1506932014Non-uniform continuity of the flow map for an evolution equation modeling shallow water waves of moderate amplitude
https://ddd.uab.cat/record/150692
We prove that the flow map associated to a model equation for surface waves of moderate amplitude in shallow water is not uniformly continuous in the Sobolev space Hs with s > 3/2. The main idea is to consider two suitable sequences of smooth initial data whose difference converges to zero in Hs, but such that neither of them is convergent. Our main theorem shows that the exact solutions corresponding to these sequences of data are uniformly bounded in Hs on a uniform existence interval, but the difference of the two solution sequences is bounded away from zero in Hs at any positive time in this interval. The result is obtained by approximating the solutions corresponding to these initial data by explicit formulae and by estimating the approximation error in suitable Sobolev norms. Duruk-Mutlubaş, NilayFri, 06 May 2016 08:59:50 GMThttps://ddd.uab.cat/record/1506922014Orbital stability of solitary waves of moderate amplitude in shallow water
https://ddd.uab.cat/record/150573
We study the orbital stability of solitary traveling wave solutions of an equation for surface water waves of moderate amplitude in the shallow water regime. Our approach is based on a method proposed by Grillakis, Shatah and Strauss in 1987 [1], and relies on a reformulation of the evolution equation in Hamiltonian form. We deduce stability of solitary waves by proving the convexity of a scalar function, which is based on two nonlinear functionals that are preserved under the flow. Duruk-Mutlubaş, NilayFri, 06 May 2016 08:54:13 GMThttps://ddd.uab.cat/record/1505732013Solitary traveling waves of moderate amplitude
https://ddd.uab.cat/record/150480
Geyer, AnnaFri, 06 May 2016 08:49:20 GMThttps://ddd.uab.cat/record/1504802012A note on uniqueness and compact support of solutions in a recent model for tsunami background flows
https://ddd.uab.cat/record/150479
Geyer, AnnaFri, 06 May 2016 08:49:20 GMThttps://ddd.uab.cat/record/1504792012On some background flows for tsunami waves
https://ddd.uab.cat/record/150478
Geyer, AnnaFri, 06 May 2016 08:49:20 GMThttps://ddd.uab.cat/record/1504782012On the wave length of smooth periodic traveling waves of the Camassa-Holm equation
https://ddd.uab.cat/record/145296
This paper is concerned with the wave length of smooth periodic traveling wave solutions of the Camassa-Holm equation. The set of these solutions can be parametrized using the wave height a (or ''peak-to-peak amplitude''). Our main result establishes monotonicity properties of the map a (a), i. e. , the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated to the equation, which distinguish between the two possible qualitative behaviours of (a), namely monotonicity and unimodality. The key point is to relate (a) to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems. Geyer, AnnaTue, 12 Jan 2016 16:38:07 GMThttps://ddd.uab.cat/record/1452962015