Topic
Cnoidal wave
About: Cnoidal wave is a research topic. Over the lifetime, 1757 publications have been published within this topic receiving 48700 citations.
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TL;DR: In this paper, it was shown that Δu=F(u) possesses non-trivial solutions in R n which are exponentially small at infinity, for a large class of functionsF. Each of them provides a solitary wave of the nonlinear Klein-Gordon equation.
Abstract: The elliptic equation Δu=F(u) possesses non-trivial solutions inR n which are exponentially small at infinity, for a large class of functionsF. Each of them provides a solitary wave of the nonlinear Klein-Gordon equation.
1,812 citations
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TL;DR: In this article, a method for obtaining traveling-wave solutions of nonlinear wave equations that are essentially of a localized nature is proposed based on the fact that most solutions are functions of a hyperbolic tangent.
Abstract: A method is proposed for obtaining traveling‐wave solutions of nonlinear wave equations that are essentially of a localized nature. It is based on the fact that most solutions are functions of a hyperbolic tangent. This technique is straightforward to use and only minimal algebra is needed to find these solutions. The method is applied to selected cases.
1,394 citations
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TL;DR: In this article, a Jacobi elliptic function expansion method was proposed to construct the exact periodic solutions of nonlinear wave equations, which includes some shock wave solutions and solitary wave solutions.
1,231 citations
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TL;DR: In this article, it is shown that advantage of crystal symmetry can be taken to construct wave functions which are best described as the smooth part of symmetrized Bloch functions.
Abstract: For metals and semiconductors the calculation of crystal wave functions is simplest in a plane wave representation. However, in order to obtain rapid convergence it is necessary that the valence electron wave functions be made orthogonal to the core wave functions. Herring satisfied this requirement by choosing as basis functions "orthogonalized plane waves." It is here shown that advantage can be taken of crystal symmetry to construct wave functions ${\ensuremath{\phi}}_{\ensuremath{\alpha}}$ which are best described as the smooth part of symmetrized Bloch functions. The wave equation satisfied by ${\ensuremath{\phi}}_{\ensuremath{\alpha}}$ contains an additional term of simple character which corresponds to the usual complicated orthogonalization terms and has a simple physical interpretation as an effective repulsive potential. Qualitative estimates of this potential in analytic form are presented. Several examples are worked out which display the cancellation between attractive and repulsive potentials in the core region which is responsible for rapid convergence of orthogonalized plane wave calculations for $s$ states; the slower convergence of $p$ states is also explained. The formalism developed here can also be regarded as a rigorous formulation of the "empirical potential" approach within the one-electron framework; the present results are compared with previous approaches. The method can be applied equally well to the calculation of wave functions in molecules.
921 citations
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TL;DR: In this article, the solitary wave solutions of the approximate equations for long water waves, coupled KdV equations and the dispersive long wave equations in 2 + 1 dimensions are constructed by using a homogeneous balance method.
865 citations