Evidence for the Nonintegrability of a Water Wave Equation in 2+1 Dimensions

2004 ◽  
Vol 59 (10) ◽  
pp. 640-644 ◽  
Author(s):  
P. R. Gordoa ◽  
A. Pickering ◽  
M. Senthilvelan
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Water Waves ◽  
Water Wave ◽  
Classification Scheme ◽  
Time Transformation ◽  
Series Solutions ◽  
Holm Equation ◽  
Proposed Model ◽  
Nonlinear Time Transformation

We provide evidence of the nonintegrability of a recently proposed model for water waves in 2+1 dimensions: we show that under a nonlinear time transformation, a certain reduction of this partial differential equation is mapped to an ordinary differential equation which does not have the Painlev´e property. This is in contrast to what happens in the case of the Camassa-Holm equation. Also, and again in contrast to the case of the Camassa-Holm equation, the equation under study fails to admit Dirichlet series solutions. - MSC2000 classification scheme numbers: 35Q51, 35Q58, 37K10.

scholarly journals A Note on Solitary Wave Solutions of the Nonlinear Generalized Camassa-Holm Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Lei Zhang ◽  
Xing Tao Wang
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Solitary Wave ◽  
Numerical Simulations ◽  
Solitary Wave Solutions ◽  
Simple Method ◽  
Wave Solutions ◽  
Holm Equation

We give a simple method for applying ordinary differential equation to solve the nonlinear generalized Camassa-Holm equation ut+2kux−uxxt+aumux−2uxuxx+uuxxx=0. Furthermore we give a new ansätz. In the cases where m=1,2,3, the numerical simulations demonstrate the results.

Partial reflexion of water waves by non-uniform adverse currents

1979 ◽  
Vol 92 (1) ◽  
pp. 119-129 ◽  
Author(s):  
M. Stiassnie ◽  
G. Dagan
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Complex Plane ◽  
Water Waves ◽  
Classical Result ◽  
Wkb Method ◽  
Energy Transmission ◽  
Total Transmission ◽  
Partial Energy ◽  
Complementary Part

The propagation of linearized waves on a non-uniform slowly varying potential current is studied by converting the equations of flow into a Schrödinger ordinary differential equation in the complex plane. This equation, which is solved by the WKB method, indicates the existence of current barriers which allow partial energy transmission while reflecting the complementary part. The classical result of total transmission (Longuet-Higgins & Stewart 1961) as well as that of complete reflexion (Peregrine 1976) are recovered as limiting cases by the present, more general approach.

VI.—Neumann-Series Solutions of the Ellipsoidal Wave Equation

1965 ◽  
Vol 67 (1) ◽  
pp. 69-77
Author(s):  
F. M. Arscott
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Wave Equation ◽  
Helmholtz Equation ◽  
Bessel Functions ◽  
Neumann Series ◽  
Series Solutions

SynopsisThe ellipsoidal wave equation is the name given to the ordinary differential equation which arises when the wave equation (Helmholtz equation) is separated in ellipsoidal co-ordinates. In this paper, solutions of the equation are expressed as Neumann series (series of Bessel functions of increasing order).

scholarly journals On finite series solutions of conformable time-fractional Cahn-Allen equation

2020 ◽  
Vol 9 (1) ◽  
pp. 194-200 ◽  
Author(s):  
Asim Zafar ◽  
Hadi Rezazadeh ◽  
Khalid K. Ali
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Fractional Order ◽  
Symbolic Computation ◽  
Nonlinear Ordinary Differential Equation ◽  
Hyperbolic Function ◽  
Series Solutions ◽  
Finite Series ◽  
New Exact Solutions

AbstractThe aim of this article is to derive new exact solutions of conformable time-fractional Cahn-Allen equation. We have achieved this aim by hyperbolic function and expa function methods with the aid of symbolic computation using Mathematica. This idea seems to be very easy to employ with reliable results. The time fractional Cahn-Allen equation is reduced to respective nonlinear ordinary differential equation of fractional order. Also, we have depicted graphically the constructed solutions.

scholarly journals Camassa-Holm equation on shallow water wave equation

2017 ◽  
Vol 5 (12) ◽  
pp. 7758-7764
Author(s):  
Sh. Hajrulla, L. Bezati, F. Hoxha
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Weak Solution ◽  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Kdv Equation ◽  
Shallow Water Wave ◽  
Holm Equation ◽  
The Cauchy Problem

In this paper we can consider the problem of week solutions for the general shallow water wave equation. In the first part of this paper, we deal to the well-known Kdv equation. We obtain the Camassa-Holm equation in particular. Both of them describe unidirectional shallow water waves equation. Moreover, all these equations have a bi-Hamiltonian structure, they are completely integrable, they have infinitely many conserved quantities. From a mathematical point of view the Camassa-Holm equation is well studied. In the second part of this paper, we obtain a global weak solution as a limit of approximation under the assumption  Some concepts related to high dimensional spaces are considered. Then the Cauchy problem is considered. It has an admissible weak solution  to the Cauchy problem for  Existence, uniqueness, and basic energy estimate on this approximate solution sequence are given in some lemmas. Finally, the main theorem and the proof is given

scholarly journals Fractional Differential Equation-Based Instantaneous Frequency Estimation for Signal Reconstruction

2021 ◽  
Vol 5 (3) ◽  
pp. 83
Author(s):  
Bilgi Görkem Yazgaç ◽  
Mürvet Kırcı
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Frequency Estimation ◽  
Signal Reconstruction ◽  
Reconstruction Method ◽  
Reconstruction Algorithms ◽  
Instantaneous Frequency Estimation ◽  
Proposed Model ◽  
Fractional Differential ◽  
Instantaneous Frequencies

In this paper, we propose a fractional differential equation (FDE)-based approach for the estimation of instantaneous frequencies for windowed signals as a part of signal reconstruction. This approach is based on modeling bandpass filter results around the peaks of a windowed signal as fractional differential equations and linking differ-integrator parameters, thereby determining the long-range dependence on estimated instantaneous frequencies. We investigated the performance of the proposed approach with two evaluation measures and compared it to a benchmark noniterative signal reconstruction method (SPSI). The comparison was provided with different overlap parameters to investigate the performance of the proposed model concerning resolution. An additional comparison was provided by applying the proposed method and benchmark method outputs to iterative signal reconstruction algorithms. The proposed FDE method received better evaluation results in high resolution for the noniterative case and comparable results with SPSI with an increasing iteration number of iterative methods, regardless of the overlap parameter.

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