Toda Molecule Equations

The main goal of this article is to find the approximate solution of the space-time fractional order KdV (STFKdV) equation and Coupled KdV (STFCKdV) equations by using Homotopy analysis method (HAM). A comparison of the exact solutions and approximate solutions has been shown from where we can say HAM is very effective and convenient for solving non linear fractional differential equation (NLFDE).

An application of the decomposition method for the generalized KdV and RLW equations

In this paper, an analytical and numerical computation of multi-solitons in Korteweg-de Vries (KdV) equation is presented. The KdV equation, which is classic of all model equations of nonlinear waves in the soliton phenomena, is described. In the analytical computation, the multi-solitons in KdV equation are computed symbolically using computer symbolic manipulator—Wolfram Mathematica via Hirota method because of the lengthy algebraic computation in the method. For the numerical computation, Crank-Nicolson implicit scheme is used to obtain numerical algorithm for the KdV equation. The simulations of solitons in MATLAB as well as results concerning collision or interactions between solitons are presented. Comparing the analytical and numerical solutions, it is observed that the results are identically equal with little ripples in solitons after a collision in the numerical simulations; however there is no significant effect to cause a change in their properties. This supports the existence of solitons solutions and the theoretical assertion that solitons indeed collide with one another and come out without change of properties or identities.

The excitation of nonlinear electron-acoustic solitary waves (SWs) is studied theoretically in an unmagnetized, uniform, and collisionless plasma. The plasma is composed of two types of electrons referred to as inertial cold electrons and inertialess Kappa (κ) distributed superthermal hot electrons along with stationary ions. Two parameters κ and β are introduced in the governing equations, where the density related parameter is defined by β = n co /n ho (n co and n ho are the cold and hot electron number densities, respectively). Using a reductive perturbation technique, nonlinear Korteweg-de Vries (KdV), modified KdV (mKdV), and standard Gardner (sG) equations have been derived, which admit SW solutions. The KdV equation allows to analyze both compressive (negative) and rarefactive (positive) solitons. But a significant solitary structure vanishes around a critical region corresponding to critical κ c and β c . To look a solitary structure in the critical region, it is proceeded to the next approach of mKdV solution with more higher order calculation of perturbation. But only rarefactive soliton is found in this approach. Furthermore, having higher order perturbation, the next approach of sG equation allows to rescue both compressive and rarefactive solitons even in the critical region. It is observed that the excitation of soliton pulses with various shapes depends on both parameters κ and β.

The dynamics of localized waves is analyzed in the framework of a model described by the Korteweg-de Vries (KdV) equation with account made for the cubic positive nonlinearity (the Gardner equation). In particular, the interaction process of two solitons is considered, and the dynamics of a "breathing" wave packet (a breather) is discussed. It is shown that solitons of the same polarity interact as in the case of the Korteweg-de Vries equation or modified Korteweg-de Vries equation, whereas the interaction of solitons of different polarity is qualitatively different from the classical case. An example of "unpredictable" behavior of the breather of the Gardner equation is discussed.

Overview of the Kadomtsev–Petviashvili-hierarchy reduction method for solitons

The Hirota bilinear method is extended to find one- and two-hump exact bright and dark soliton solutions to a coupled system between the linear Schrödinger and Korteweg-de Vries (KdV) equations arising in the energy transfer problem along a cubic anharmonic crystal medium. The bilinear form associated to this system is found by imitating the well known bilinearizing transformations used in the standard nonlinear Schrödinger (NLS) and KdV equations. Proper finite exponential expansions in the transformed variables allow one to exhibit multihump soliton solutions as single entities resulting from the adjustment of appropriate dispersion relations between the wave parameters describing the profiles. It is found that such one- and two-hump solutions correspond to the one- and two-KdV solitons trapped by both the bright and dark-gray NLS solitons. Our two-hump bright and dark solutions represent novel solutions for the type of interactions and nonlinearities considered.

In this paper, we determine spectral stability results of periodic waves for the critical Korteweg-de Vries and Gardner equations. For the first equation, we show that both positive and zero mean periodic traveling wave solutions possess a threshold value which may provides us a rupture in the spectral stability. Concerning the second equation, we establish the existence of periodic waves using a Galilean transformation on the periodic cnoidal solution for the modified Korteweg-de Vries equation and for both equations, the threshold values are the same. The main advantage presented in our paper concerns in solving some auxiliary initial value problems to obtain the spectral stability.

CONTENTS Introduction Chapter I. Theta functions. General information § 1. Definition of theta functions and their simplest properties § 2. Theta functions of a single variable § 3. On Abelian tori § 4. Addition theorems for theta functions Chapter II. Theta functions of Riemann surfaces. The Jacobi inversion problem § 1. Periods of Abelian differentials on Riemann surfaces. Jacobi varieties § 2. Abel's theorem § 3. Some remarks on divisors on a Riemann surface § 4. The Jacobi inversion problem. Examples Chapter III. The Baker-Akhiezer function. Applications to non-linear equations § 1. The Baker-Akhiezer one-point function. The Kadomtsev-Petviashvili equation and equations associated with it § 2. The Baker-Akhiezer two-point function. The Schrödinger equation in a magnetic field Chapter IV. Effectivization of the formulae for the solution of KdV and KP equations. Recovery of a Riemann surface from its Jacobi variety. The problem of Riemann and the conjecture of Novikov § 1. The KdV equation. Genus or 2 § 2. The KP equation. Genus 2 and 3 § 3. The KP equation. Genus . Canonical equations of Riemann surfaces § 4. The problem of Riemann on relations between the periods of holomorphic differentials on a Riemann surface and the conjecture of Novikov Chapter V. Examples of Hamiltonian systems that are integrable in terms of two-dimensional theta functions § 1. Two-zone potentials § 2. The problem of Sophie Kovalevskaya § 3. The problems of Neumann and Jacobi. The general Garnier system § 4. Movement of a solid in an ideal fluid. Integration of the Clebsch case. A multi-dimensional solid Appendix. I.M. Krichever. The periodic non-Abelian Toda chain and its two-dimensional generalization References

Solitary wave solutions for the KdV and mKdV equations by differential transform method

Multi-scale Runge-Kutta_Galerkin method for solving one-dimensional KdV and Burgers equations

I. SOLITON ON A SLOPING BEACH AND RELATED PROBLEMS The problem of the behaviour of a soliton on a slowly varying beach is considered. It is shown that for a correct description, the full Boussinesq equations rather than a Korteweg-de Vries type approximation must be used. Using both energy conservation and two-timing expansions, the behaviour of the soliton is analysed. The slowly varying soliton is found not to conserve mass and momentum and it has been suggested that to conserve these quantities, both forward and reflected waves must be added behind the soliton, these waves being solutions of the linear shallow water equations. It is shown that to the order of approximation of the Boussinesq equations, only a forward wave (or tail) behind the soliton is necessary to fulfill mass and momentum conservation. A perturbed Korteweg-de Vries equation for which the perturbation adds energy to the soliton is considered. It is found that a tail is formed behind the soliton. The development of this tail into new solitons is analysed. II. MODULATED CAPILLARY WAVES An exact hodograph solution for symmetric and antisymmetric capillary waves on a fluid sheet (of possibly infinite thickness) has been previously found. Using this solution, an exact averaged Lagrangian for slowly varying capillary waves is calculated. Modulation equations can be found from this averaged Lagrangian, but due to the algebraic complexity of the equations, the limit of waves on a thin fluid sheet is considered. From the modulation equations, the stability of symmetric and antisymmetric capillary waves on a thin fluid sheet is found. The modulation equations for antisymmetric waves form a hyperbolic system and the simple wave solutions for this system are calculated. These simple wave solutions are interpreted physically.

The classical problem of irrotational long waves on the surface of a shallow layer of an ideal fluid moving under the influence of gravity as well as surface tension is considered. A systematic procedure for deriving an equation for surface elevation for a prescribed relation between the orders of the two expansion parameters, the amplitude parameter α and the long wavelength (or shallowness) parameter β, is developed. Unlike the heuristic approaches found in the literature, when modifications are made in the equation for surface elevation itself, the procedure starts from the consistently truncated asymptotic expansions for unidirectional waves, a counterpart of the Boussinesq system of equations for the surface elevation and the bottom velocity, from which the leading-order and higher order equations for the surface elevation can be obtained by iterations. The relations between the orders of the two small parameters are taken in the form β = O(αn) and α = O(βm) with n and m specified to some important particular cases. The analysis shows, in particular, that some evolution equations, proposed before as model equations in other physical contexts (such as the Gardner equation, the modified Korteweg–de Vries (KdV) equation and the so-called fifth-order KdV equation), can emerge as the leading-order equations in the asymptotic expansion for the unidirectional water waves, on equal footing with the KdV equation. The results related to the higher orders of approximation provide a set of consistent higher order model equations for unidirectional water waves which replace the KdV equation with higher order corrections in the case of non-standard ordering when the parameters α and β are not of the same order of magnitude. The shortcomings of certain models used in the literature become apparent as a result of the subsequent analysis. It is also shown that various model equations obtained by assuming a prescribed relation β = O(αn) between the orders of the two small parameters can be equivalently treated as obtained by applying transformations of variables which scale out the parameter β, in favor of α. It allows us to consider the nonlinearity-dispersion balance, epitomized by the soliton equations, as existing for any β, provided that α → 0, but leads to a prescription, in asymptotic terms, of the region of time and space where the equations are valid and so the corresponding dynamics are expected to occur.

The main purpose of the paper is to provide a survey of our recent studies on soliton solutions of the Kadomtsev–Petviashvili (KP) equation. The KP equation describes weakly dispersive and small amplitude wave propagation in a quasi-two-dimensional framework. Recently, a large variety of exact soliton solutions of the KP equation has been found and classified. These solutions are localized along certain lines in a two-dimensional plane and decay exponentially everywhere else, and are called line-solitons. The classification is based on the far-field patterns of the solutions which consist of a finite number of line-solitons. Each soliton solution is then defined by a point of the totally non-negative Grassmann variety which can be parametrized by a unique derangement of the symmetric group of permutations. Our study also includes certain numerical stability problems of those soliton solutions. Numerical simulations of the initial value problems indicate that certain classes of initial waves asymptotically approach to these exact solutions of the KP equation. We discuss an application of our theory to the Mach reflection problem in shallow water. This problem describes the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall, and it predicts an extraordinary fourfold amplification of the wave at the wall. There are several numerical studies confirming the prediction, but all indicate disagreements with the KP theory. Contrary to those previous numerical studies, we find that the KP theory actually provides an excellent model to describe the Mach reflection phenomena when the higher order corrections are included in the quasi-two-dimensional approximation. We also present laboratory experiments of the Mach reflection recently carried out by Yeh and his colleagues, and show how precisely the KP theory predicts this wave behavior.

In this paper, a new generalized algebraic method is proposed to construct a series of explicit exact solutions of general nonlinear evolution equations with variable coefficients. Compared with most existing methods, such as the tanh method, the extended tanh method, the Jacobi elliptic function method or the algebraic method, the proposed method seems to be more powerful. The efficiency of the method is demonstrated by applying it to the (1+1)-dimensional variable coefficients modified Korteweg-de Vries (MKdV) equation and the (2+1)-dimensional variable coefficients Kadomtsev- Petviashvili (KP) equation. A rich variety of new exact explicit solutions has been found. - PACS numbers: 03.40.Kf; 03.65.Fd; 02.30Jr.