modified Korteweg-de Vries equation

Optical solitons are localized electromagnetic waves that propagate steadily in a nonlinear medium resulting from the robust balance between nonlinearity and linear broadening due to dispersion and diffraction. Existence of the optical soliton was first time found in 1973 when Hasegawa and Tappert demonstrated the propagation of a pulse through a nonlinear optical fiber described by the nonlinear Schr¨odinger equation [8]. They performed a number of com- puter simulations demonstrating that nonlinear pulse transmission in optical fibers would be stable. Subsequently, after the fabrication of low-loss fiber, Mollenauer et al. in 1980 success- fully confirmed this theoretical prediction of soliton propagation in a laboratory experiment [7]. Since then, fiber solitons have emerged as a very promising potential candidate in long-haul fiber optic communication systems.

considered the existence and uniqueness of global solutions to the Cauchy problem related to this equation [6, 18, 19, 24, 40]. Moreover, He et al. [20] considered the KS equation when the boundary conditions are periodic and proved the stability of this equation. In [14], the authors proved that the mixed problem for the KS equation is globally well-posed in a bounded domain with moving boundaries. They also proved the exponential decay of the solutions with L 2 -norm. In [24], the author considered the generalized Kuramoto- Sivashinsky equation (i.e., when α = γ 1 = 1, μ > 0, and γ 2 = –σ > 0 in (1.1)). He proved the

Fractional calculus has been used in almost all fields of engineering, mathematics and other natural sciences. Recently, several mathematical methods have been applied to PDEs. Two-dimensional differential transform method was implemented to obtain ap- proximate analytical solutions of fractional modified Korteweg-de Vries(fmKdV) equa- tion [1]. Homotopy analysis method and its modification were used to solve fmKdV equation [2]. Similarly the homotopy-perturbation method was also capable of get- ting approximate analytical solution of fmKdV equation [3]. In [4], the travelling wave solutions were expressed in terms of hyperbolic, trigonometric and rational functions. Then (G 0 /G)-expansion method was applied for the analytical solutions of the fmKdV equations. Exact solutions of the fmKdV equation were investigated by the generalized Kudryashov method by Bulut et al. [5]. In that paper, soliton solutions and hyperbolic function solutions were constructed by using the properties of exponential functions.

t = 5 seconds are presented by parallel lines with a negative slope less than –ν = –0.01, and this is in accordance with the analytical results given by inequality (50). Therefore, one can conclude from Figs. 4–6 that the L 2 -norm, u(x, t), converges exponentially to zero as t tends to infinity. However, it should be noted that depending whether α is odd or even as it increases, the solutions of the MGKdVB equation takes longer time to reach the steady state solution.

Indeed, an electron-positron plasma usually behaves as a fully ionized gas consisting of electrons and positrons, as seen in the solar atmo- sphere as well as in many astrophysical objects (e.g., white dwarfs, neu- tron stars, near the polar cap of pulsars, in the active galactic nuclei, in the early universe). The success achieved in collecting and keep- ing positrons and even anti-hydrogen under laboratory conditions has opened up a new field of laboratory anti-matter plasma. The propaga- tion ion-acoustic waves is one the most important subjects in plasma physics and the study of ion acoustic waves in plasmas has received con- siderable attention, because its key role in understanding the nonlinear- phenomena in laboratory plasmas [13] as well as in space plasmas [14, 19]. There are usually, two methods for investigation ion-acoustic waves in plasmas; one of them is Sagdeev pseudo-potential method [15], where in this method the main properties of arbitrary amplitude ionacous- tic waves by obtain an integral energy equation is studied. Another method which is used for studies nonlinear phenomena in plasmas is the reductive perturbation method [11]. In this method nonlinear waves in plasma can be described by different partial differential equations as Korteweg-de-Vries equation [8], modified Kortewegde-Vries (mKdV) equation [18] and the nonlinear Schrodinger equation (NLSE) [6, 17] investigated effects of ion temperature on ion-acoustic waves in a non- thermal plasmas, by using the pseudo-potential approach, which is valid for arbitrary amplitude solitary waves. It has shown that to increase ion temperature the amplitude of the compressive and rarefactive solitary wave’s decreases. Linear and nonlinear properties and the modulation of ion-acoustic waves in plasmas with two temperature electrons which having different Boltzmann distributions theoretically, investigated by several authors with different method[5].

from the literature that as an important nonlin- ear topic, nowadays, the solitary waves are be- ing studied extensively both theoretically and ex- perimentally. It is so because in various fields of science and engineering, nonlinear evaluation equations, as well as their analytic and numer- ical solutions, are of fundamental importance. One of the most attractive and surprising wave phenomenon is the creation of solitary waves or solitons. Solitons are self-localized wave packets arising from a robust balance between dispersion and nonlinearity. In small amplitude approxima- tion, one ends up deriving some forms of nonlin- ear differential equations like Korteweg-de Vries (KdV) or modified Korteweg-de Vries (mKdV) or nonlinear Schrodinger equation, etc. which have solitary or solitonic solutions. It was ap-

2003), periodic solutions (Gesztesy & Holden 2003), traveling-waves (Parkes & Vakhnenko 2005), (Lenells 2005a). The construction of multi-soliton and multi-positon solutions for the Associated Camassa-Holm equation using the Darboux/B¨acklund transform is presented in (Schiff 1998), (Hone 1999) and (Ivanov, 2005b). The N -soliton solution for the DP equation was recently derived by Matsuno (2005a), the multi-peakon solutions for DP are obtained by Lundmark & Szmigielski (2003, 2005); the traveling waves – by Parkes & Vakhnenko (2004) and Lenells (2005b). There are similarities between the CH and DP equations, in a sense that they both are integrable and have a hydrodynamic derivation. However, it is interesting to notice that only CH has a geometric interpretation as a geodesic flow, cf. (Constantin & Kolev 2003), (Kolev 2004).

where n is a positive integer. Motivated by the results obtained by Chen and Wen [2], the author look for the real-valued traveling wave solutions of the form U(ξ) = u(x,y,z,t) with ξ = ax + by + cz − ωt, where a, b, c, and ω are real constants. Without loss of generality, we can assume a > 0. Substituting the U(ξ) into (2.1), we are then led to look for solutions of the following fourth order ordinary differential equation

Nouri and Sloan [9] studied six Fourier pseudo-spectral methods that solve the KdV equation numerically namely leap-frog scheme of Fornberg and Witham, semi-implicit scheme of Chan and Kerhoven, modified basis function scheme of Chan and Kerhoven, split-step scheme based on Taylor expansion, split-step scheme based on characteristics and quasi-Newton implicit method. They found that the semi-implicit scheme of Chan and Kerhoven [4] to be the most efficient of the methods tested. Chan and Kerhoven integrated the KdV equation in time in Fourier space using two Fast Fourier Transform (FFT) per time step. They also used Crank-Nicolson method for the linear term and a leap-frog method for the nonlinear term.

respectively. One can see that the first three equations are elliptic Diophantine equa- tions, thus using the program package MAGMA, subroutines IntegralPoints or IntegralQuarticPoints are just a straightforward calculation to solve them. In these cases, the unique solution is (l, m, n) = (, , ). The forth equation can be written as follows

The Korteweg-de Vries-Burgers equation is one of the typical examples incorporating the effects of dispersion, dissipation and nonlinearity. This model provides a discription of the propagation of waves on an elastic tube filled with a viscous fluid [1], and also describes the phenomenon of containing turbulence [2] and gas bubbles [3]. Anzayko et al. have investigated the ferroelectricity problem of the Korteweg-de Vries-Burgers equation [4, 5]. Hayashi et al., [6] gave the local existence of the solution of the initial-boudnary value problem of KDV-B equations on half-line, and they also showed global existence and asymptotic behavior in time of solutions under sufficient conditions accordingly. Guo and Wang also have proved the global well-posedness and inviscid limit for the KDV-B equation [7]. Molinet and Vento have got the sharp ill-posedness and well-posedness result for the KDV-B equation on the one dimension torus T = R/2πZ [8].

Mainly, this research will discuss more about the multi-solitons solutions of KdV equation up to six-solitons. The Hirota‟s bilinear method will be used to obtain these solutions of KdV equation. Soliton or also known as solitary wave growth in broad field such as shallow and deep water waves, fibre optics, protein and DNA, magnet, bions, and biological models.

The MOL is a powerful method used to solve partial differential equations (PDEs). It involves making an approximation to the spatial derivatives and reducing the problem into a system of ODEs (Hall & Watt, 1976; Loeb & Schiesser, 1974; Schiesser, 1994). In addition, this system of ODEs can be solved by using time integrator. The most important advantage of the MOL approach is that it has not only the simplicity of the explicit methods (Dehgan, 2006) but also the superiority (stability advantage) of the implicit ones unless a poor numerical method for solution of ODEs is employed. It is possible to achieve higher-order approximations in the discretization of spatial derivatives without significant increases in the computational complexity. This method has wide applicability to physical and chemical systems modeled by PDEs such as delay differential equations (Koto, 2004), two-dimensional sine-Gordon equation (Bratsos, 2007), the Nwogu one-dimensional extended Boussinesq equation(Hamdi et.al, 2005),the fourth-order Boussinesq equation, the fifth-order Kaup–Kupershmidt equation and an extended Fifth- Order Korteweg-de Vries (KdV5) equation(Saucez et al, 2004).

u x t for Equation (12), with α = 0.01 , β = − 5 , γ = 3 , κ = 1 . The range of x and t are x ∈ − [ 10,10 ] and t ∈ − [ 10,10 ] , which is shown in Figure 4. From the figures, we know that the amplitude of the traveling wave solutions changes with time. All figures are smooth and no singularity in the given time and given field. These solutions help us find the peak and the deep location in physics system.

Recently, there has been interest in the fact that a particular generalization of the classical korteweg-de Vries (KdV) equation can support a new type of solitary wave, which has been referred to as a compacton in the literature. This type of wave has a compact support and a width which is independent of the amplitude of the wave [11, 12]. This equation is defined by the real parameters (m,n) and given in terms of the function u(x,t) by the fully nonlinear KdV equation [4]

solutions. We have to produce the permutation parameters of solitons interactions in order to obtain all the eight soliton solutions of KdV equation. As these solutions are difficult to calculate manually, thus we need a computer programming tools to derive the function and produce the various graphical outputs for up to eight-soliton solutions of

With the rapid development of science and technology, the study kernel of modern science is changed from lin- ear to nonlinear step by step. Many nonlinear science problems can simply and exactly be described by using the mathematical model of nonlinear equation. Up to now, many important physical nonlinear evolution equa- tions are found, such as sin-Gordon equation, KdV equa- tions, Schrodinger equation all possess solitary wave solutions. There exist many methods to seek for the soli- tary wave solutions, such as inverse scattering method, Hopf-Cole transformation, Miura transformations, Dar- boux transformation and Bäcklund transformation [2-5], but solving nonlinear equations is still an important task. In this paper, with the aid of Mathematica, a traveling

Let us now recall the original motivation to consider PT -symmetrically extended models, which was to exploit the feature that unbroken PT -symmetry guarantees the reality of the corresponding spectrum. In this spirit it is highly desirable to discriminate between the models, which are Hamiltonian systems and those which are not. It is well known that the sKdV-equation admits a Hamiltonian description for λ = 2, see [18], and it is interesting to investigate whether this feature survives the deformation.

We shall now construct solutions of the equations of motion (3.5). One may expect to find a rich variety of different types of solutions similarly as for the standard KdV equation. Over the years several methods have been developed to find such solutions ranging from minimizing the sum of the conserved charges [18], the inverse scattering method [19], Hirota’s bilinearization method [20], etc. Some methods demand as a prerequisite the model to be integrable. As this feature is not guaranteed for the model at hand, in fact the conjecture is that the model is not integrable, our aim is here just to obtain a first impression in order to indicate that the above family of equations deserve further attention. Following a simple procedure which has turned out to be useful for the standard KdV equation, we may integrate (3.5) directly by assuming the solution to be a steady progressing wave

D 3 + uD where D = d/dx, such knowledge is needed for the main results. In Section 3, we reduce the existence of delay KdV-type equation (1.1) to the local existence of mild solution of abstract integral equation. In Section 4, we will investigate the classical solution of (1.1) and some remarks.