Modified Kortweg-de Vries Research Articles

Nonlinear Wave Propagation Through Cold Plasma

Electromagnetic wave propagation through cold collision free plasma is studied using the nonlinear perturbation method. It is found that the equations can be reduced to the modified Kortweg-de Vries equation.

Effects of vortex-like and non-thermal ion distributions on non-linear dust-acoustic waves

The effects of vortex-like and non-thermal ion distributions are incorporated in the study of nonlinear dust-acoustic waves in an unmagnetized dusty plasma. It is found that owing to the departure from the Boltzmann ion distribution to a vortex-like phase space distribution, the dynamics of small but finite amplitude dust-acoustic waves is governed by a modified Kortweg–de Vries equation. The latter admits a stationary dust-acoustic solitary wave solution, which has larger amplitude, smaller width, and higher propagation velocity than that involving adiabatic ions. On the other hand, consideration of a non-thermal ion distribution provides the possibility of coexistence of large amplitude rarefactive as well as compressive dust-acoustic solitary waves, whereas these structures appear independently when the wave amplitudes become infinitely small. The present investigation should help us to understand the salient features of the non-linear dust-acoustic waves that have been observed in a recent numerical simulation study.

Intercomparison of Truncated Series Solutions for Shallow Water Waves

The relationship between truncated Fourier series solutions of several different long-wave evolution equations is explored. The models are all designed to represent the same physics in the asymptote of small nonlinearity and frequency dispersion, and yet give different numerical solutions for given values of dimensionless parameters. It is found that the problem of obtaining a steady solution in a coupled-mode model of shallow-water wave evolution is related more to the problem of properly choosing the corresponding time-dependent evolution equation than to the problem of truncating the infinite series representation of the solution to that equation. In the process, a particular lowest-order coupled-mode model is related to a particular modified form of the Kortweg–de Vries equation. It is shown that the existing lowest-order model may be inherently inaccurate in the case of relatively high waves, due to the nonphysical behavior of the form of the nonlinear term inherent in that model. Finally, solitary and cnoidal wave solutions are given for the entire family of modified Kortweg–de Vries equations, and their properties are compared.

The quantum inverse method and the monodromy matrix in integrable field theories

Under investigation in this paper is a variable-coefficient modified Kortweg-de Vries (vc-mKdV) model describing certain situations from the fluid mechanics, ocean dynamics and plasma physics. N-fold Darboux transformation (DT) of a variable-coefficient Ablowitz–Kaup–Newell–Segur spectral problem is constructed via a gauge transformation. Multi-solitonic solutions in terms of the double Wronskian for the vc-mKdV model are derived by the reduction of the N-fold DT. Three types of the solitonic interactions are discussed through figures: (1) Overtaking collision; (2) Head-on collision; (3) Parallel solitons. Nonlinear, dispersive and dissipative terms have the effects on the velocities of the solitonic waves while the amplitudes of the waves depend on the perturbation term.

The Bäcklund trasnformation for the integrable many-body systems

Under investigation in this paper is a variable-coefficient modified Kortweg-de Vries (vc-mKdV) model describing certain situations from the fluid mechanics, ocean dynamics and plasma physics. N-fold Darboux transformation (DT) of a variable-coefficient Ablowitz–Kaup–Newell–Segur spectral problem is constructed via a gauge transformation. Multi-solitonic solutions in terms of the double Wronskian for the vc-mKdV model are derived by the reduction of the N-fold DT. Three types of the solitonic interactions are discussed through figures: (1) Overtaking collision; (2) Head-on collision; (3) Parallel solitons. Nonlinear, dispersive and dissipative terms have the effects on the velocities of the solitonic waves while the amplitudes of the waves depend on the perturbation term.

Connection between the infinite sequence of Lie–Bäcklund symmetries of the Korteweg–de Vries and sine-Gordon equations

From the observation that the infinite sequence of Lie–Bäcklund symmetries of the potential modified Kortweg–deVries (PMK–dV) and the sine–Gordon (s–G) equations are identical, it is shown that there exists a simple connection between the Lie–Bäcklund symmetries (written in the form of evolution equations) of the Korteweg–deVries (K–dV) and s–G equations. Further, this connection is similar to the one obtained by Chodos for the conserved quantities of K–dV and s–G equations. We also point out that the result of Chodos can be realized from the equality of conserved densities of PMK–dV and s–G systems.

Propagation of Drift Waves of Small but Finite Amplitude

It is shown that the drift wave of small but finite amplitude propagating in a plasma of hot electrons and cold ions is governed by a modified Kortweg-de Vries equation in two-dimensional space comprising the external magnetic field and the direction normal to a density gradient. The solitary wave is obtained explicitly.