2000
DOI: 10.1007/pl00001508
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Abstract: In the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thin tube and the approximate equations of an incompressible inviscid fluid, the propagation of weakly nonlinear waves, in such a medium is studied through the use of the modified multiple scale expansion method. It is shown that the evolution of the lowest order (first order) term in the perturbation expansion may be described by the Korteweg-de Vries equation. The governing equation for the second order terms a… Show more

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Cited by 12 publications
(10 citation statements)
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“…To eliminate such secularities various methods, like the renormalization method of Kodama and Taniuti [4] and the multiple scale expansion method by Kraenkel and Manna [6], have been presented in the current literature. The results of the present work and of those given in [12] and [13] proved that the "modified reductive perturbation method", presented by us, is the most simple and effective one. The present problem has been studied by Kraenkel et al [14] but the method they used is quite complicated compared to ours.…”
Section: Resultssupporting
confidence: 67%
See 1 more Smart Citation
“…To eliminate such secularities various methods, like the renormalization method of Kodama and Taniuti [4] and the multiple scale expansion method by Kraenkel and Manna [6], have been presented in the current literature. The results of the present work and of those given in [12] and [13] proved that the "modified reductive perturbation method", presented by us, is the most simple and effective one. The present problem has been studied by Kraenkel et al [14] but the method they used is quite complicated compared to ours.…”
Section: Resultssupporting
confidence: 67%
“…However, this approach can only be used when one studies progressive wave solutions to the original nonlinear equations, and it does not give any idea about the form of evolution equations governing the various order terms in the perturbation expansion. In our previous paper [12], we have presented the so-called "modified reductive perturbation method" to examine the contributions of higher-order terms in the perturbation expansion and applied it to weakly dispersive ion-acoustic plasma waves and solitary waves in a fluid-filled elas-tic tube [13]. In these works, we have shown that the lowest-order term in the perturbation expansion is governed by the nonlinear Korteweg -de Vries equation, whereas the higher-order terms in the expansion are governed by the degenerate Korteweg -De Vries equation with the nonhomogeneous term.…”
Section: Introductionmentioning
confidence: 99%
“…However, this approach can only be used when one studies progressive wave solution to the original nonlinear equations and it does not give any idea about the form of evolution equations governing the various order terms in the perturbation expansion. In our previous paper [10], a method so called "the modified reductive perturbation method" had been presented to examine the contributions of higher order terms in the perturbation expansion and applied it to weakly dispersive ion-acoustic plasma waves and solitary waves in a fluid filled elastic tube [11], In these works, it is shown that the lowest order term in the perturbation expansion is governed by the nonlinear Korteweg-de Vries equation, whereas the higher order terms in the expansion are governed by the degenerate Korteweg-de Vries equation with nonhomogeneous term. By employing hyperbolic tangent method a progressive wave type of solution was sought and the possible secularities were removed by selecting the scaling parameter in a special way.…”
Section: Introductionmentioning
confidence: 99%
“…However, this approach can only be used when one studies the progressive wave solution to the original nonlinear equations and it does not give any idea about the form of evolution equations governing the various order terms in the perturbation expansion. In our previous paper [8], we have presented a method called ''the modified reductive perturbation method'' to examine the contributions of higher order terms in the perturbation expansion and applied it to weakly dispersive ion-acoustic plasma waves, solitary waves in a fluid-filled elastic tube [9] and long water waves of constant depth [10]. In these works, we have shown that the lowest order term in the perturbation expansion is governed by the nonlinear Korteweg-de Vries equation, whereas the higher order terms in the expansion are governed by the degenerate Korteweg-de Vries equation with the nonhomogeneous term.…”
Section: Introductionmentioning
confidence: 99%