2,434 research outputs found
Some New Exact Solutions of Jacobian Elliptic Functions in Nonlinear Physics Problem
Abstract:An extended mapping method with symbolic computation is developed to obtain some new periodic wave solutions in terms of Jacobin elliptic function for nonlinear elastic rod equation arising in physics.As a result,many exact travelling wave solutions are obtained which include Jacobian elliptic functions solutions,combined Jacobian elliptic functions solutions and triangular function solutions.Solutions in the limiting cases have also been studied.It is shown that the mapping method provides a very effective and powerful mathematical tool for solving nonlinear evolution equations in physics
On Boussinesq-type models for long longitudinal waves in elastic rods
In this paper we revisit the derivations of model equations describing long
nonlinear longitudinal bulk strain waves in elastic rods within the scope of
the Murnaghan model in order to derive a Boussinesq-type model, and extend
these derivations to include axially symmetric loading on the lateral boundary
surface, and longitudinal pre-stretch. We systematically derive two forced
Boussinesq-type models from the full equations of motion and non-zero surface
boundary conditions, utilising the presence of two small parameters
characterising the smallness of the wave amplitude and the long wavelength
compared to the radius of the waveguide. We compare the basic dynamical
properties of both models (linear dispersion curves and solitary wave
solutions). We also briefly describe the laboratory experiments on generation
of bulk strain solitary waves in the Ioffe Institute, and suggest that this
generation process can be modelled using the derived equations.Comment: 19 pages, 5 figures, submitted to the Special Issue of Wave Motion,
"Nonlinear Waves in Solids", in Memory of Professor Alexander M. Samsono
Solitary smooth hump solutions of the Camassa-Holm equation by means of the homotopy analysis method
The homotopy analysis method is used to find a family of solitary smooth hump solutions of the Camassa-Holm equation. This approximate solution, which is obtained as a series of exponentials, agrees well with the known exact solution. This paper complements the work of Wu & Liao [Wu W, Liao S. Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos, Solitons & Fractals 2005;26:177-85] who used the homotopy analysis method to find a different family of solitary wave solutions
Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions
In this article, we investigate two types of double dispersion equations in two different dimensions, which arise in several physical applications. Double dispersion equations are derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Firstly, we obtain conservation laws for both these equations. To do this, we employ the multiplier method, which is an efficient method to derive conservation laws as it does not require the PDEs to admit a variational principle. Secondly, we obtain travelling waves and line travelling waves for these two equations. In this process, the conservation laws are used to obtain a triple reduction. Finally, a line soliton solution is found for the double dispersion equation in two dimensions
Swirling fluid flow in flexible, expandable elastic tubes: variational approach, reductions and integrability
Many engineering and physiological applications deal with situations when a
fluid is moving in flexible tubes with elastic walls. In the real-life
applications like blood flow, there is often an additional complexity of
vorticity being present in the fluid. We present a theory for the dynamics of
interaction of fluids and structures. The equations are derived using the
variational principle, with the incompressibility constraint of the fluid
giving rise to a pressure-like term. In order to connect this work with the
previous literature, we consider the case of inextensible and unshearable tube
with a straight centerline. In the absence of vorticity, our model reduces to
previous models considered in the literature, yielding the equations of
conservation of fluid momentum, wall momentum and the fluid volume. We show
that even when the vorticity is present, but is kept at a constant value, the
case of an inextensible, unshearable and straight tube with elastics walls
carrying a fluid allows an alternative formulation, reducing to a single
compact equation for the back-to-labels map instead of three conservation
equations. That single equation shows interesting instability in solutions when
the vorticity exceeds a certain threshold. Furthermore, the equation in stable
regime can be reduced to Boussinesq-type, KdV and Monge-Amp\`ere equations
equations in several appropriate limits, namely, the first two in the limit of
long time and length scales and the third one in the additional limit of the
small cross-sectional area. For the unstable regime, we numerical solutions
demonstrate the spontaneous appearance of large oscillations in the
cross-sectional area.Comment: 57 pages, 11 figures. arXiv admin note: text overlap with
arXiv:1805.1102
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