Modified Korteweg-de Vries Soliton Research Articles

Modified Korteweg–de Vries solitons with quartic nonlinearity in a dusty plasma

The present multicomponent dusty plasma with ions, Cairns distributed electrons and immobile dusts has been investigated first time through modified Korteweg–de Vries (mKdV) equation of quartic nonlinearity derived by reductive perturbation technique. In this new investigation, it is found that the dust ion-acoustic (DIA) solitary waves have smaller amplitudes compared to the amplitudes of mKdV-DIA compressive solitons of our previous investigation []. Roles of non-thermal parameter (β), dust to ion density ratio (σ), number of dust charge (Z d ) and initial streaming speed of ion (u i0) in the growth of amplitudes and widths of this new mKdV-DIA solitons are investigated.

Comparative study of dust ion acoustic Korteweg–de Vries and modified Korteweg–de Vries solitons in dusty plasmas with variable temperatures

In this plasma model, consisting of ions and electrons with pressure variations in both the components in the presence of stationary dust, both compressive and rarefactive Korteweg–de Vries (KdV) solitons of interesting character are established. Based on high dust charge, characteristics of soliton growth are found to be amplified for various pairs of ion and electron streaming speeds. It is noteworthy to mention that for some pairs of ion and electron initial streaming speeds, only compressive KdV solitons with either decreasing or increasing growth are shown to reflect. Contrary to this, for some other pairs of ion and electron streaming speeds, the amplitudes of both rarefactive and compressive solitons are seen to be produced, changing from rarefactive to compressive growth. At the stationary background of the massive dust particles, the lighter particles suffer appreciable initial drifts (backwards streaming) which characteristically change the growth of solitons. For inclusion of higher-order nonlinearity, only compressive modified Korteweg–de Vries (MKdV) solitons of much higher amplitude are found to exist whereas for the same set of parameter values both compressive and rarefactive KdV solitons are found to exist. Smaller values of electron streaming speed are seen to produce high amplitude MKdV solitons. We also observe that due to higher-order nonlinearity, the nonlinear monotonic growth of amplitudes of MKdV solitons is supported by the almost equal streaming speed pairs of ions and electrons for relatively small values of$Z_{d}$, where$Z_{d}$is the number of charges in a dust particle

Modified Korteweg–de Vries solitons at supercritical densities in two-electron temperature plasmas

The supercritical composition of a plasma model with cold positive ions in the presence of a two-temperature electron population is investigated, initially by a reductive perturbation approach, under the combined requirements that there be neither quadratic nor cubic nonlinearities in the evolution equation. This leads to a unique choice for the set of compositional parameters and a modified Korteweg–de Vries equation (mKdV) with a quartic nonlinear term. The conclusions about its one-soliton solution and integrability will also be valid for more complicated plasma compositions. Only three polynomial conservation laws can be obtained. The mKdV equation with quartic nonlinearity is not completely integrable, thus precluding the existence of multi-soliton solutions. Next, the full Sagdeev pseudopotential method has been applied and this allows for a detailed comparison with the reductive perturbation results. This comparison shows that the mKdV solitons have slightly larger amplitudes and widths than those obtained from the more complete Sagdeev solution and that only slightly superacoustic mKdV solitons have acceptable amplitudes and widths, in the light of the full solutions.

Linear and nonlinear obliquely propagating ion-acoustic waves in magnetized negative ion plasma with non-thermal electrons

AbstractIon-acoustic solitons in magnetized low-β plasma consisting of warm adiabatic positive and negative ions and non-thermal electrons have been studied. The reductive perturbation method is used to derive the Korteweg–de Vries (KdV) equation for the system, which admits an obliquely propagating soliton solution. It is found that due to the presence of finite ion temperature there exist two modes of propagation, namely fast and slow ion-acoustic modes. In the case of slow-mode if the ratio of temperature to mass of positive ion species is lower (higher) than the negative ion species, then there exist compressive (rarefactive) ion-acoustic solitons. It is also found that in the case of slow mode, on increasing the non-thermal parameter (γ) the amplitude of the compressive (rarefactive) soliton decreases (increases). In fast ion-acoustic mode the nature and characteristics of solitons depend on negative ion concentration. Numerical investigation in case of fast mode reveals that on increasing γ, the amplitude of compressive (rarefactive) soliton increases (decreases). The width of solitons increases with an increase in non-thermal parameters in both the modes for compressive as well as rarefactive solitons. There exists a value of critical negative ion concentration (αc), at which both compressive and rarefactive ion-acoustic solitons appear as described by modified KdV soliton. The value of αc decreases with increase in γ.

Transition of ion-acoustic perturbations in multicomponent plasma with negative ions

Evolution of ion-acoustic compressive (positive) and rarefactive (negative) perturbations in a multicomponent plasma with negative ions has been investigated in a double plasma device. Transition of compressive solitons in electron-positive ion plasma, into a dispersing train of oscillations in a multicomponent plasma, when the negative ion concentration r exceeds a critical value rc, has been observed. On the other hand, an initial rarefactive perturbation initially evolves into a dispersing train of oscillations in electron-positive ion plasma and transforms into rarefactive solitons in a multicomponent plasma when the negative ion concentration is higher than the critical value. The Mach velocity and width of the compressive and rarefactive solitons are measured. The compressive solitons in the range 0<r<rc and the rarefactive solitons in the range r>rc have different characteristics than the Korteweg–de Vries (KdV) solitons at r=0 and modified KdV solitons at r=rc. A nonlinear differential equation having two terms to account for the lower and higher order nonlinearity has been used to explain the observed results.

Soliton propagation in an inhomogeneous plasma at critical density of negative ions: Effects of gyratory and thermal motions of ions

The effects of gyratory and thermal motions of ions on soliton propagation in an inhomogeneous plasma that contains positive ions, negative ions, and electrons are studied at a critical density of negative ions. Since at this critical negative ion density the nonlinear term of the relevant Korteweg–deVries (KdV) equation vanishes, a higher order of nonlinearity is considered by retaining higher-order perturbation terms in the expansion of dependent quantities together with the appropriate set of stretched coordinates. Under this situation, time-dependent perturbation leads to the evolution of modified KdV solitons, which are governed by a modified form of the KdV equation that has an additional term due to the density gradient present in the plasma. On the basis of the solution of this equation and obliquely applied magnetic field, the effects of gyratory and thermal motions of ions are analyzed on the soliton propagation for three cases, nn0<ne0, nn0=ne0, and nn0>ne0, together with nn0 (ne0) as the density of negative ions (electrons). The role of the negative ions in the evolution of the modes and the solitons is also discussed. Under the limiting cases, our calculations reduce to the ones obtained by other investigators in the past. This substantiates the generality of the present analysis.

A comparative study of modified Korteweg–de Vries (MKdV) and Korteweg–de Vries (KdV) solitons in plasmas with negative ions under the influence of electrons’ drift motion

In the plasma under consideration, the existence of modified Korteweg–de Vries (MKdV) solitons has been established for different values of electrons’ drift velocities νe′ and of Q′ (=mj/mi, negative to positive ion mass ratio). The amplitudes of the MKdV solitons are found to be much smaller (<1) than those of the corresponding amplitudes (>1) of the KdV solitons for Q′<0.6 when νe′=10. Unlike the existence of both compressive and rarefactive KdV solitons with two unaccountable regions of νe′, there is found to exist only compressive MKdV solitons of small amplitudes in the lower but entire range of νe′ for U=1.09, 1.12, and 1.15 when Q′=0.4. It is noteworthy to mention that, the width of the KdV soliton does not depend on Q′, i.e., on the concentration of negative ions whereas that of the MKdV soliton depends on Q′.

PHYSICAL REALIZATION OF THE JIMBO-MIWA THEORY OF THE MODIFIED KORTEWEG-DE VRIES EQUATION ON A THIN ELASTIC ROD: FERMIONIC THEORY

Recently we found that the Dirac operator on a thin elastic rod is identical with the Lax operator of the modified Korteweg-de Vries (MKdV) equation while the thin elastic rod is governed by the MKdV equation. In this article, we will show the physical relation between the Hirota bilinear method and the Dirac field in a thin rod on two-dimensional space, along the lines of the Jimbo-Miwa construction of the MKdV soliton.

The Relation between the Modified Korteweg-de Vries Equation and an Anomaly of the Dirac Field on a Thin Elastic Rod

Recently a study on the Dirac equation in a thin elastic rod as a submanifold quantum system was reported. (S. Matsutani and H. Tsuru, Phys. Rev. A46 (1992), 1144). It is shown that the Dirac operator is identified with the Lax operator of the Modified Korteweg-de Vries (MKdV) equation. Since the thin elastic rod is governed by the MKdV equation, it implies that the fictitious quantum mechanics in the soliton physics has a real physical meaning as a fermionic field on the elastic rod. In this paper, we quantize the Dirac field in a thin elastic rod on R • We find that there appears an anomalous current proportional to the curvature of the rod which is the MKdV soliton. Hence we prove that the relation between the fermionic field systell). and the classical MKdV soliton is regarded as an anomaly on the thin elastic rod.

Modified Korteweg–deVries solitons in a warm plasma with negative ions

The propagation of ion-acoustic solitons in a warm plasma in the presence of negative ions has been investigated theoretically through the modified Korteweg–deVries (KdV) equation. In the case of the first-order velocity perturbation of the positive ions bearing a constant ratio to that of the negative ions, the existence of both compressive and rarefactive solitons at different critical densities for different temperature ratios has been established when the mass ratio is greater than unity. It is also demonstrated in this case that no modified KdV soliton exists at the critical density in a warm plasma with negative ions when the mass ratio is less than unity.

A Nonlinear Normal Mode in an Anharmonic Lattice with a Single Defect

A nonlinear wave propagating along a one-dimensional anharmonic lattice with a single defect is investigated theoretically by using a continuum approximation. The Modified Korteweg-de Vries (MK-dV) equation is shown by applying the reductive perturbation method to this lattice, and a one soliton is obtained as its stationary solution. It is newly shown that the amplitude, the velocity and the nonlinear coupling coefficient of the soliton vary according to a coordinate moving at the velocity of wave. The MK-dV soliton obtained in this paper accounts for a nonlinear normal mode of vibration in a one-dimensional anharmonic lattice system containing a single defect.