Solitary waves in an elastic rod: analytical solutions

Dinh, T. Bui; Long, Van Cao; Leoński, W.

doi:10.1117/12.2006521

Cited by 1 publication

(3 citation statements)

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“…In Ref. some exact analytical soliton solutions were obtained which will be compared with the numerical results obtained here. For this purpose Eq.…”

Section: Analytical Solutions Of the Double Dispersion Equationmentioning

confidence: 93%

“…(11), the left‐hand sides of Eq. (11) are converted into polynomials of, then setting each coefficient to zero, one obtains a system of algebraic equations for b 0 , b 1 , b 2 , b 3 , b 4 , k , A 0 , and c . Solving this system by MAPLE one obtains

\begin{array}{l} center & b_{0} = - \frac{4 k^{2} Q (- a_{2} c^{2} + a_{3}) + 1 - c^{2}}{2 a_{1}}, b_{1} = b_{3} = b_{4} = 0, b_{2} = \frac{6 k^{2} P (- a_{2} c^{2} + a_{3})}{a_{1}}, \end{array}

where k and c are arbitrary non‐zero constants.…”

Section: Analytical Solutions Of the Double Dispersion Equationmentioning

confidence: 99%

“…It is well‐known that existence of solitary waves in a nonlinear system results from a balance between nonlinearity and dispersion. As it has been analyzed in , nonlinearity in a pure rod is caused by a finite stress values and the elastic material properties while dispersion results from the finite transverse size of the rod. The leading nonlinearity for longitudinal waves is quadratic which leads to the double dispersion equation (DDE) of Porubov's type .…”

Section: Introductionmentioning

confidence: 99%

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Solitary waves in auxetic rods with quadratic nonlinearity: Exact analytical solutions and numerical simulations

Dinh

Long

Wojciechowski

2015

Physica Status Solidi (b)

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Special soliton analytical solutions are seeked for the so-called double dispersion equation (or Porubov's equation), which describes the propagation of longitudinal waves in an elastic rod. Using the F-expansion method, a new interesting class of traveling solitary waves is obtained. As a byproduct, the results obtained by other authors for some special cases are regained. Some numerical simulations are also performed. Splitting of various initial pulses during propagation into a sequence of solitary waves is considered. The dependence of the amplitude and the velocity of the solitary waves on Poisson's ratio is discussed in detail. Collisions between some obtained solitary waves are also presented. The simulation results are compared with the obtained exact analytical solutions-the latter reflect the perfect balance between the nonlinearity and dispersion. The comparison indicates that the obtained exact solutions are useful for describing more complicated wave fields studied by numerical simulations. It also indicates that the numerical results for auxetic materials are in better agreement with theoretical solutions than in the case of ordinary materials. Furthermore, one can conclude that the Secant pulse generates solitary waves closer to the analytical predictions than a Gaussian pulse. This is similar to the case of variational method, where the Secant trial function is also more proper than the Gaussian one.

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“…In Ref. some exact analytical soliton solutions were obtained which will be compared with the numerical results obtained here. For this purpose Eq.…”

Section: Analytical Solutions Of the Double Dispersion Equationmentioning

confidence: 93%