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Application of higher-order KdV——mKdV model with higher-degree nonlinear terms to gravity waves in atmosphere
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Higher-order Korteweg-de Vries (KdV)-modified KdV (mKdV) equations
with a higher-degree of nonlinear terms are derived from a simple
incompressible non-hydrostatic Boussinesq equation set in atmosphere
and are used to investigate gravity waves in atmosphere. By taking
advantage of the auxiliary nonlinear ordinary differential equation,
periodic wave and solitary wave solutions of the fifth-order
KdV--mKdV models with higher-degree nonlinear terms are obtained
under some constraint conditions. The analysis shows that the
propagation and the periodic structures of gravity waves depend on
the properties of the slope of line of constant phase and atmospheric
stability. The Jacobi elliptic function wave and solitary wave
solutions with slowly varying amplitude are transformed into
triangular waves with the abruptly varying amplitude and breaking
gravity waves under the effect of atmospheric instability. 相似文献
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The method of multiple-scales is used to investigate the evolution of a weak nonlinear internal waves between two-layer fluids in cylindrical coordinate system. Two reduced model wave equations, which we call a modified cylindrical KdV equation for axially symmetric case and a modified cylindrical KP equation for non-axially symmetric case, are derived and their solitary wave solutions are also obtained by relating them i to the modified KdV equation by means of an appropriate variable transformation. 相似文献
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《Journal of Nonlinear Mathematical Physics》2013,20(1-2):139-146
Abstract An efficient method for constructing of particular solutions of some nonlinear partial differential equations is introduced. The method can be applied to nonintegrable equations as well as to integrable ones. Examples include multisoliton and periodic solutions of the famous integrable evolution equation (KdV) and the new solutions, describing interaction of solitary waves of nonintegrable equation. 相似文献
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D. Ch. Kim 《Technical Physics》2013,58(3):340-350
Possible types of low-frequency electromagnetic solitary waves in a dispersive LC transmission line with a quadratic or cubic capacitive nonlinearity are investigated. The fourth-order nonlinear wave equation with ohmic losses is derived from the differential-difference equations of the discrete line in the continuum approximation. For a zero-loss line, this equation can be reduced to the nonlinear equation for a transmission line, the double dispersion equation, the Boussinesq equations, the Korteweg-de Vries (KdV) equation, and the modified KdV equation. Solitary waves in a transmission line with dispersion and dissipation are considered. 相似文献
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Consistent Riccati expansion solvability,symmetries, and analytic solutions of a forced variable-coefficient extended Korteveg-de Vries equation in fluid dynamics of internal solitary waves
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We study a forced variable-coefficient extended Korteweg-de Vries (KdV) equation in fluid dynamics with respect to internal solitary wave. Bäcklund transformations of the forced variable-coefficient extended KdV equation are demonstrated with the help of truncated Painlevé expansion. When the variable coefficients are time-periodic, the wave function evolves periodically over time. Symmetry calculation shows that the forced variable-coefficient extended KdV equation is invariant under the Galilean transformations and the scaling transformations. One-parameter group transformations and one-parameter subgroup invariant solutions are presented. Cnoidal wave solutions and solitary wave solutions of the forced variable-coefficient extended KdV equation are obtained by means of function expansion method. The consistent Riccati expansion (CRE) solvability of the forced variable-coefficient extended KdV equation is proved by means of CRE. Interaction phenomenon between cnoidal waves and solitary waves can be observed. Besides, the interaction waveform changes with the parameters. When the variable parameters are functions of time, the interaction waveform will be not regular and smooth. 相似文献
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DUAN Wen-Shan 《理论物理通讯》2004,41(1):115-118
In a thin-walled, homogeneous, straight, long, circular, and
incompressible fluid filled elastic tube, small but finite long
wavelength nonlinear waves can be describe by a KdV (Korteweg de
Vries) equation, while the carrier wave modulations are described
by a nonlinear Schrödinger equation (NLSE). However if the elastic
tube is slowly inhomogeneous, then it is found, in this paper, that
the carrier wave
modulations are described by an NLSE-like equation. There are
soliton-like solutions for them, but the stability and instability regions for this
soliton-like waves will change, depending on what
kind of inhomogeneity the tube has. 相似文献
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Bifurcation and solitary waves of the nonlinear wave equation with quartic polynomial potential
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For the nonlinear wave equation with quartic polynomial potential, bifurcation and solitary waves are investigated. Based on the bifurcation and the energy integral of the two-dimensional dynamical system satisfied by the travelling waves, it is very interesting to find different sufficient and necessary conditions in terms of the bifurcation parameter for the existence and coexistence of bright, dark solitary waves and shock waves. The method of direct integration is developed to give all types of solitary wave solutions. Our method is simpler than other newly developed ones. Some results are similar to those obtained recently for the combined KdV-mKdV equation. 相似文献
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We propose a simple and direct method for generating travelling wave solutions for nonlinear integrable equations. We illustrate how nontrivial solutions for the KdV, the mKdV and the Boussinesq equations can be obtained from simple solutions of linear equations. We describe how using this method, a soliton solution of the KdV equation can yield soliton solutions for the mKdV as well as the Boussinesq equations. Similarly, starting with cnoidal solutions of the KdV equation, we can obtain the corresponding solutions for the mKdV as well as the Boussinesq equations. Simple solutions of linear equations can also lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve some new families of KdV equations and show how soliton solutions are also obtained for the higher order equations of the KdV hierarchy using this method. 相似文献
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《Waves in Random and Complex Media》2013,23(1):56-76
Abstract In this paper, we introduce and study rigorously a Hamiltonian structure and soliton solutions for a weakly dissipative and weakly nonlinear medium that governs two Korteweg–de vries (KdV) wave modes. The bounded solution and traveling wave solution such as cnoidal wave and solitary wave are obtained. Subsequently, the equation is numerically solved by Fourier spectral method for its two-soliton solution. These solutions may be useful to explain the nonlinear dynamics of waves for an equation supporting multi-mode weakly dispersive and nonlinear wave medium. In addition, we give an explicit explanation of the mathematics behind the soliton phenomenon for a better understanding of the equation. 相似文献
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An extended functional transformation method and its application in some evolution equations 总被引:1,自引:0,他引:1
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In this paper, an extended functional transformation is given to solve some nonlinear evolution equations. This function, in fact,is a solution of the famous KdV equation, so this transformation
gives a transformation between KdV equation and other soliton equations. Then many new exact solutions can be given by virtue of the solutions of KdV equation. 相似文献
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The dynamics of the weak nonlinear matter solitary waves in a spin-1condensates with harmonic external potential are investigated analytically by a perturbation method. It is shown that, in the small amplitude limit, the dynamics of the solitary waves are governed by a variable-coefficient Korteweg-de Vries (KdV) equation. The reduction to the (KdV) equation may be useful tounderstand the dynamics of nonlinear matter waves in spinor BECs. The analytical expressions for the evolution of soliton show that the small-amplitude vector solitons of the mixed types perform harmonic oscillations in the presence of the trap. Furthermore, the emitted radiation profiles and the soliton oscillation frequency are also obtained. 相似文献
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Ghosh S. Sarkar S. Khan M. Gupta M.R. 《IEEE transactions on plasma science. IEEE Nuclear and Plasma Sciences Society》2001,29(3):409-416
Some properties of nonlinear dust acoustic waves in magnetized dusty plasma with variable charges by reductive perturbation technique have been studied. The effect of adiabatic dust charge variations under the assumption that the ratio of dust charging time to the dust hydrodynamical time is very small, and the nonadiabatic dust charges variations under the assumption that the same ratio is small but finite, are also incorporated. It is seen that the magnetic field and the dust charge variations significantly modify the wave amplitude. It is also seen that in case of adiabatic charge variations, the Korteweg-de Vries (KdV) equation governs the nonlinear dust acoustic wave, whereas in case of nonadiabatic dust charge variations, the wave is governed by the KdV Burger equation. Nonadiabaticity generated anomalous dissipative effect causes generation of the dust acoustic shock wave. Numerical integration of KdV Burger equation shows that the dust acoustic wave admits oscillatory (dispersion dominant) or monotone (dissipation dominant) shock solutions depending on the magnitude of the coefficient of the Burger term 相似文献
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从理论上研究了非热离子、外部磁场、碰撞对非均匀热尘埃等离子体中三维非线性尘埃声孤波的影响。运用约化摄动法得到描述三维非线性尘埃声孤波的非标准的变系数Korteweg-de Vries(KdV)方程。然后把非标准KdV方程变为标准的变系数KdV方程,并且得到了标准的变系数KdV方程的近似解析解。由此解析解可以看出,非热离子的数目、碰撞、非均匀性、波的斜向传播、尘埃颗粒和非热离子的温度对三维非线性尘埃声孤波的振幅和宽度有很大的影响。外部磁场对三维非线性尘埃声孤波的宽度有影响,而对其振幅没有影响。此外,波的相速度与非热离子、波的斜向传播、尘埃颗粒的温度和非均匀性有关。 相似文献
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A general mapping approach and new travelling wave solutions to the general variable coefficient KdV equation
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A general mapping deformation method is applied to a generalized variable coefficient KdV equation. Many new types of exact solutions, including solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions and other exact excitations are obtained by the use of a simple algebraic transformation relation between the generalized variable coefficient KdV equation and a generalized cubic nonlinear Klein-Gordon equation. 相似文献
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Vladimirov S.V. Yu M.Y. 《IEEE transactions on plasma science. IEEE Nuclear and Plasma Sciences Society》1993,21(2):250-253
A simple model for studying finite-amplitude ionizing nonlinear surface waves propagating in a partially ionized low-temperature plasma, in which collisional effects such as ionization, recombination, and friction are dominant, is proposed. The authors consider the lowest order namely, second order in the fields) nonlinear problem and investigate the evolution of finite-amplitude electromagnetic surface waves. It is shown that the waves are governed by a modified Korteweg-de Vries (KdV) equation and that new types of solitary waves can exist. The structures of the latter are mathematically similar to, but physically quite different from, that of the KdV soliton 相似文献
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A kind of extended Korteweg——de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system
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This paper considers interfacial waves propagating along the
interface between a two-dimensional two-fluid with a flat bottom and
a rigid upper boundary. There is a light fluid layer overlying a
heavier one in the system, and a small density difference exists
between the two layers. It just focuses on the weakly non-linear
small amplitude waves by introducing two small independent
parameters: the nonlinearity ratio $\varepsilon $, represented by
the ratio of amplitude to depth, and the dispersion ratio $\mu $,
represented by the square of the ratio of depth to wave length,
which quantify the relative importance of nonlinearity and
dispersion. It derives an extended KdV equation of the interfacial
waves using the method adopted by Dullin {\it et al} in the study of
the surface waves when considering the order up to $O(\mu ^2)$. As
expected, the equation derived from the present work includes, as
special cases, those obtained by Dullin {\it et al} for surface
waves when the surface tension is neglected. The equation derived
using an alternative method here is the same as the equation
presented by Choi and Camassa. Also it solves the equation by
borrowing the method presented by Marchant used for surface waves,
and obtains its asymptotic solitary wave solutions when the weakly
nonlinear and weakly dispersive terms are balanced in the extended
KdV equation. 相似文献
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研究了小的有限振幅的无磁场尘埃等离子体中的非线性波.在一维情况下由Kortewegde Veries(KdV)方程来描述,考虑了二维情况下尘埃等离子体中尘埃颗粒上电荷的变化效应以及双温度离子效应后,尘埃等离子体受到横向高阶扰动后动力学方程由Kadomtsev-Petviashvili(KP)方程来描述.在此基础上,研究了以任意夹角传播的两个及三个孤立子的相互作用问题,考虑非线性效应后振幅相等的双孤立子在相互作用区域内振幅最大值是单个孤立子振幅的4倍,振幅相等的三孤立子在相互作用区域内振幅最大值是单个孤立子振幅的9倍.研究还表明波的传播方向受到横向高阶扰动后是稳定的. 相似文献