mKdV Type Research Articles

New unexpected variety of solitons arising from spatio-temporal dispersion (1+1)-dimensional Ito-equation

In this work, new varieties of unexpected scenarios for the solitons arising from the spatio-temporal dispersion ([Formula: see text])-dimensional Ito-equation (STDIE) are considered as an extension of the KdV (mKdV) type to higher order and usually employed to predict the rolling behavior of ships in regular sea. Moreover, they can describe the interaction process of two internal long waves specially the height of the water’s free surface above a flat bottom where [Formula: see text] is an analytic function. We will introduce these new unexpected and impressive scenarios for the solitons that emerged from this model for the first time in the framework of three distinct schemas. The three schemas that are employed for this target are the Paul-Painleve approach method (PPAM), the extended direct algebraic method (EDAM) and ([Formula: see text]/G)-expansion method. Our obtained solitons are new compared with that achieved before by other authors who applied various schemas.

Hamiltonian approach to modelling interfacial internal waves over variable bottom

We study the effects of an uneven bottom on the internal wave propagation in the presence of stratification and underlying non-uniform currents. Thus, the presented models incorporate vorticity (wave–current interactions), geophysical effects (Coriolis force) and a variable bathymetry. An example of the physical situation described above is well illustrated by the equatorial internal waves in the presence of the Equatorial Undercurrent (EUC). We find that the interface (physically coinciding with the thermocline and the pycnocline) satisfies in the long wave approximation a KdV–mKdV type equation with variable coefficients. The soliton propagation over variable depth leads to effects such as soliton fission, which is analysed and studied numerically as well.

On local well-posedness and ill-posedness results for a coupled system of mkdv type equations

<p style='text-indent:20px;'>We consider the initial value problem associated to a coupled system of modified Korteweg-de Vries type equations <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} \partial_tv + \partial_x^3v + \partial_x(vw^2) = 0,&amp;v(x,0) = \phi(x),\\ \partial_tw + \alpha\partial_x^3w + \partial_x(v^2w) = 0,&amp; w(x,0) = \psi(x), \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>and prove the local well-posedness results for a given data in low regularity Sobolev spaces <inline-formula><tex-math id="M1">\begin{document}$ H^{s}( \rm{I}\! \rm{R})\times H^{k}( \rm{I}\! \rm{R}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ s,k&gt; -\frac12 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ |s-k|\leq 1/2 $\end{document}</tex-math></inline-formula>, for <inline-formula><tex-math id="M4">\begin{document}$ \alpha\neq 0,1 $\end{document}</tex-math></inline-formula>. Also, we prove that: (I) the solution mapping that takes initial data to the solution fails to be <inline-formula><tex-math id="M5">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula> at the origin, when <inline-formula><tex-math id="M6">\begin{document}$ s&lt;-1/2 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M7">\begin{document}$ k&lt;-1/2 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M8">\begin{document}$ |s-k|&gt;2 $\end{document}</tex-math></inline-formula>; (II) the trilinear estimates used in the proof of the local well-posedness theorem fail to hold when (a) <inline-formula><tex-math id="M9">\begin{document}$ s-2k&gt;1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M10">\begin{document}$ k&lt;-1/2 $\end{document}</tex-math></inline-formula> (b) <inline-formula><tex-math id="M11">\begin{document}$ k-2s&gt;1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ s&lt;-1/2 $\end{document}</tex-math></inline-formula>; (c) <inline-formula><tex-math id="M13">\begin{document}$ s = k = -1/2 $\end{document}</tex-math></inline-formula>;

On the existence and formation of small amplitude electrostatic double‐layer structure in nonthermal dusty plasma

AbstractIn the present article, we studied the effect of nonthermal electrons on the formation and existence of double‐layer structures in a three‐species plasma consisting of positive ions, nonthermal electrons, and immobile negative dust‐charged grains. Employing the reductive perturbations, a modified Korteweg–de Vries (mKdV) type equation is derived for the dust‐ion‐acoustic waves (DIAWs) bearing nonthermality. We found that both positive and negative polarity shock structures (double layer) can exist such that it switches polarity while changing the dust charge concentrations. However, strong nonthemality favours only rarefactive structures irrespective of the ion temperature. It is also found that increasing the nonthermal electron in the system the width of the double layer is increased; furthermore, the shock structure forms with small dust charge concentration. For small ionic temperature, increasing the nonthermal electrons in the system makes the double layer potential to increase; however, for σ = 1 reverse phenomena occurs. Our results are relevant to the shock observations in Q machine experiments and in the ionospheric regime of the earth.

A remark on the local well-posedness for a coupled system of mKdV type equations in H^s × H^k

A remark on the local well-posedness for a coupled system of mKdV type equations in H^s × H^k

The Sylvester Equation and Integrable Equations: The Ablowitz—Kaup—Newell—Segur System

In this paper, we seek connections between the Sylvester equation and the Ablowitz—Kaup—Newell—Segur (AKNS) system. By the Sylvester equation KM — MK = r sT, we introduce master function S(i, j) = sT Kj (I + M)−1Kir. This function satisfies some recurrence relations. By imposing dispersion relations on r and s, we study the constructions of the AKNS system, where some AKNS type equations are investigated emphatically, including second-AKNS equation, second-modified AKNS (mAKNS) equation, third-AKNS equation, third-mAKNS equation and (—1)st-AKNS equation. The reductions of these equations to complex Korteweg—de Vries (KdV) equation, real and complex modified Korteweg—de Vries (mKdV) type equations, nonlinear Schrodinger (NLS) type equations and sine-Gordon (sG) equation are discussed.

Logarithm transformation together with the integral bifurcation method for investigating exact solutions of the CDF equation with mKdV type

In this paper, under a logarithm transformation, by using the integral bifurcation method, we study the Cologero–Degasperis–Fokas equation with mKdV type, which was introduced by Cologero, Degasperis and Fokas. Some exact traveling wave solutions such as smooth solitary wave solutions, smooth periodic wave solutions, kink wave solutions, singular kink wave solution, singular periodic wave solutions, blowup soliton solution and singular blowup wave solution are obtained. We show their profiles and discuss their dynamic properties aim at some typical solutions.

Dust-Acoustic Solitary Waves in an Unmagnetized Dusty Plasma with Arbitrarily Charged Dust Fluid and Trapped Ion Distribution

The nonlinear propagation of dust-acoustic (DA) solitary waves in three-component unmagnetized dusty plasma consisting of Maxwellian electrons, vortex-like (trapped) ions, and arbitrarily charged cold mobile dust grain has been investigated. It has been found that, owing to the departure from the Maxwellian ions distribution to a vortex-like one, the dynamics of small but finite amplitude DA waves is governed by a nonlinear equation of modified Korteweg-de Vries (mK-dV) type instead of K-dV. The reductive perturbation method has been employed to study the basic features (phase speed, amplitude, width, etc.) of DA solitary waves which are significantly modified by the presence of trapped ions. The implications of our results in space and laboratory plasmas are briefly discussed.

New integrable models with &amp;lt;italic&amp;gt;N&amp;lt;/italic&amp;gt;-peakons and algebro-geometric solutions of soliton equations

In the soliton theory, it is a basic and challenging problem to search for integrable dynamical models with peaked soliton solutions and obtain the explicit solutions to soliton equations. This paper can be mainly divided into two parts. First, the derivation of two integrable dynamical systems with N -peakon, which provides new models for the family of nonlinear evolution equations with N -peakon; on the other hand, based on the knowledge of hyperelliptic curve, the method of nite-order expansion of Lax pairs is improved to obtain the solution of KdV6 equations. Moreover, to avoid the limit of using the Riemann theorem, the meromorphic function and the Baker-Akhiezer vector are introduced by which quasi-periodic solutions to the mKdV type equations and mixed AKNS equations are constructed according to their asymptotic properties and algebro-geometric characters.

Nonlinear propagation of dust-acoustic solitary waves in a dusty plasma with arbitrarily charged dust and trapped electrons

A theoretical investigation of dust-acoustic solitary waves in three-component unmagnetized dusty plasma consisting of trapped electrons, Maxwellian ions, and arbitrarily charged cold mobile dust was done. It has been found that, owing to the departure from the Maxwellian electron distribution to a vortex-like one, the dynamics of small but finite amplitude dust-acoustic (DA) waves is governed by a nonlinear equation of modified Korteweg–de Vries (mKdV) type (instead of KdV). The reductive perturbation method was employed to study the basic features (amplitude, width, speed, etc.) of DA solitary waves which are significantly modified by the presence of trapped electrons. The implications of our results in space and laboratory plasmas are briefly discussed.

On the conservation laws for nonlinear partial difference equations

In this paper, a direct method is proposed to construct conservation laws for nonautonomous nonlinear partial difference equations (PΔEs). The identified PΔEs can be classified into QRT type, KdV type, mKdV type and sG type. The integrability of the identified partial difference equations is also discussed.

On integrability of a [formula omitted]-dimensional MKdV-type equation

A ( 2 + 1 ) -dimensional equation of MKdV type proposed by Hietarinta is considered. It is shown that this equation has N-solition solution represented in terms of Wronski determinants. Also, a bilinear Bäcklund transformation and a Lax representation are proposed.

Hamiltonian Flows of Curves in G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type

The bi-Hamiltonian structure of the two known vector generalizations of the mKdV hierarchy of soliton equations is derived in a geometrical fashion from flows of non- stretching curves in Riemannian symmetric spaces G/SO(N). These spaces are exhausted by the Lie groups G = SO(N + 1),SU(N). The derivation of the bi-Hamiltonian structure uses a parallel frame and connection along the curve, tied to a zero curvature Maurer-Cartan form on G, and this yields the mKdV recursion operators in a geometric vectorial form. The kernel of these recursion operators is shown to yield two hyperbolic vector generalizations of the sine-Gordon equation. The corresponding geometric curve flows in the hierarchies are described in an explicit form, given by wave map equations and mKdV analogs of Schrodinger map equations.

Higher-order nonlinearity of electron-acoustic solitary waves with vortex-like electron distribution and electron beam

The nonlinear wave structure of small-amplitude electron-acoustic solitary waves (EASWs) is investigated in a four-component plasma consisting of cold electron fluid, hot electrons obeying vortex-like distribution traversed by a warm electron beam and stationary ions. The streaming velocity of the beam, uo, plays the dominant role in determining the roots of the linear dispersion relation associated with the system. Using the reductive perturbation theory, the basic set of equations is reduced to a modified Korteweg–de Vries (mKdV) equation. With the inclusion of higher-order nonlinearity, a linear inhomogeneous mKdV type equation with fifth-order dispersion term is derived and the higher-order solution is obtained using a renormalization method. However, both mKdV and mKdV-type solutions present a positive potential, which corresponds to a hole (hump) in the cold (hot) electron number density. The mKdV-type solution has a smaller energy amplitude and a wider width than that of mKdV solution. The dependence of the energy amplitude, the width, and the velocity on the system parameters is investigated. The findings of this investigation are used to interpret the electrostatic solitary waves observed by the Geotail spacecraft in the plasma sheet boundary layer of the Earth’s magnetosphere.

N=4 SUGAWARA CONSTRUCTION ON $\widehat{{\rm sl}(2|1)}$, $\widehat{{\rm sl}(3)}$ AND mKdV-TYPE SUPERHIERARCHIES

The local Sugawara constructions of the "small" N=4 SCA in terms of supercurrents of N=2 extensions of the affine [Formula: see text] and [Formula: see text] algebras are investigated. The associated super mKdV type hierarchies induced by N=4 SKdV ones are defined. In the [Formula: see text] case the existence of two nonequivalent Sugawara constructions is found. The "long" one involves all the affine [Formula: see text] currents, while the "short" one deals only with those from the subalgebra [Formula: see text]. As a consequence, the [Formula: see text]-valued affine superfields carry two nonequivalent mKdV type super-hierarchies induced by the correspondence between "small" N=4 SCA and N=4 SKdV hierarchy. However, only the first hierarchy possesses genuine global N=4 supersymmetry. We discuss peculiarities of the realization of this N=4 supersymmetry on the affine supercurrents.

Tau-functions generating the conservation laws for generalized integrable hierarchies of KdV and affine toda type

For a class of generalized integrable hierarchies associated with affine (twisted or untwisted) Kac-Moody algebras, an explicit representation of their local conserved densities by means of a single scalar tau-function is deduced. This tau-function acts as a partition function for the conserved densities, which fits its potential interpretation as the effective action of some quantum system. The class consists of multi-component generalizations of the Drinfel'd-Sokolov and the two-dimensional affine Toda lattice hierarchies. The relationship between the former and the approach of Feigin, Frenkel and Enriquez to soliton equations of KdV and mKdV type is also discussed. These results considerably simplify the calculation of the conserved charges carried by the soliton solutions to the equations of the hierarchy, which is important to establish their interpretation as particles. By way of illustration, we calculate the charges carried by a set of constrained KP solitons recently constructed.

Doubly discrete Lagrangian systems related to the Hirota and sine-Gordon equation

We extend the action for evolution equations of KdV and mKdV type which was derived by Capel and Nijhoff to the case of not periodic, but only equivariant phase space variables, introduced by Faddeev and Volkov. The difference of these variables may be interpreted as reduced phase space variables via a Marsden-Weinstein reduction where the monodromies play the role of the momentum map. As an example we obtain the doubly discrete sine-Gordon equation and the Hirota equation and the corresponding symplectic structures.

LOOP EQUATIONS AND THE TOPOLOGICAL PHASE OF MULTI-CUT MATRIX MODELS

We study the double scaling limit of mKdV type, realized in the two-cut Hermitian matrix model. Building on the work of Periwal and Shevitz and of Nappi, we find an exact solution including all odd scaling operators, in terms of a hierarchy of flows of 2×2 matrices. We derive from it loop equations which can be expressed as Virasoro constraints on the partition function. We discover a “pure topological” phase of the theory in which all correlation functions are determined by recursion relations. We also examine macroscopic loop amplitudes, which suggest a relation to 2D gravity coupled to dense polymers.

A well-defined multicritical series in hermitian one-matrix models

A new multicritical series is described for hermitian one-matrix models with even potential U( φ) = Σ d i = 1 g 2 i φ 2 i . It is characterized by a negative g 2 and a positive g 2 d . The latter condition ensures that the matrix model is well-defined. The corresponding value of γ 0 is −1/( d − 1). The specific heat is F″ = 1 4 u 2 , where u satisfies a string equation of the mKdV type. For the case d=3, a numerical solution is presented which is pole free and completely determined by asymptotic boundary conditions. This new sequence, as well as the other two sequences previously found for hermitian one-matrix models with even potentials, are shown to be special cases of a two-parameter family of multicritical models characterized by an engenvalue distribution with n ⩽ 2 cuts.

Local Hamiltonian structures of multicomponent KdV equations

The author discusses the behaviour of Hamiltonian structures for a generalised AKNS hierarchy under componentwise reductions to equations of KdV and MKdV type, identifying those cases where a new local structure arises.