Exact Periodic and Solitary-Wave Solutions of Multi-component Modified Korteweg-de Vries Equations

The present work extends the search of Jacobi elliptic function solutions for the multi-component modified Korteweg-de Vries equations. When the modulum m →1, those periodic solutions degenerate as the corresponding solitary wave and shock wave ones. Especially, exact solutions for the three-component system are presented in detail and graphically.     

Dynamics of solitons in a nonintegrable version of the modified Korteweg-de Vries equation

Nonlinear wave dynamics is discussed using the extended modified Korteweg-de Vries equation that includes the combination of the third- and fifth-order terms and is valid for waves in a three-layer fluid with so-called symmetric stratification. The derived equation has solutions in the form of solitary waves of various polarities. At small amplitudes, they are close to solitons of the modified Korteweg-de Vries equation. However, the height of large-amplitude solutions has a limit approaching which solitary waves widen and acquire a table like shape similar to soluitons of the Gardner equation. Numerical calculations confirm that the collision of solitons of the derived equation is inelastic. Inelasticity is the most pronounced in the interaction of unipolar pulses. The direction of the shift of the phase of the higher-amplitude soliton owing to the interaction of solitons of different polarities depends on the amplitudes of the pulses.     

Physical solutions for unitary matrix models

We study the simplest double scaling limit of the integral over a unitary matrix, shown by Periwal and Shevitz to admit an exact solution in terms of the mKdV hierarchy. We show that there is a unique non-perturbative solution of the string equation corresponding to the true double scaling limit of the integral, which interpolates smoothly between weak and strong coupling regimes.     

Riemann theta function periodic wave solutions for the variable-coefficient mKdV equation

<正>In this paper,a variable-coefficient modified Korteweg-de Vries(vc-mKdV) equation is considered.Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function,then the one and two periodic wave solutions are presented,and it is also shown that the soliton solutions can be reduced from the periodic wave solutions.     

Emergence and control of multiphase nonlinear waves by synchronization

Large amplitude multiphase solutions of the periodic Korteweg-de Vries equation are excited and controlled by a small forcing. The approach uses passage through an ensemble of resonances and subsequent multiphase self-locking of the system with eikonal-type perturbations. The synchronization of each phase in the Korteweg-de Vries wave is robust, provided the corresponding driving amplitude exceeds a threshold.     

Solutions to the Complex Korteweg-de Vries Equation: Blow-up Solutions and Non-Singular Solutions

In the paper two kinds of solutions are derived for the complex Korteweg-de Vries equation, includ- ing blow-up solutions and non-singular solutions. We derive blow-up solutions from known 1-soliton solution and a double-pole solution. There is a complex Miura transformation between the complex Korteweg-de Vries equation and a modified Kortcweg-de Vries equation. Using the transformation, solitons, breathers and rational solutions to the com- plex Korteweg-de Vries equation are obtained from those of the modified Korteweg-de Vries equation. Dynamics of the obtained solutions are illustrated.     

Quasi-periodic and periodic solutions for dynamical systems related to Korteweg-de Vries equation

We consider quasi-periodic and periodic (cnoidal) wave solutions of a set of n-component dynamical systems related to Korteweg-de Vries equation. Quasi-periodic wave solutions for these systems are expressed in terms of Novikov polynomials. Periodic solutions in terms of Hermite polynomials and generalized Hermite polynomials for dynamical systems related to Korteweg-de Vries equation are found. Received 15 October 2001 / Received in final form 6 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: nakostov@ie.bas.bg     

New binary travelling-wave periodic solutions for the modified KdV equation

In this Letter, the modified Korteweg-de Vries (mKdV) equations with the focusing (+) and defocusing (−) branches are investigated, respectively. Many new types of binary travelling-wave periodic solutions are obtained for the mKdV equation in terms of Jacobi elliptic functions such as sn(ξ,m)cn(ξ,m)dn(ξ,m) and their extensions. Moreover, we analyze asymptotic properties of some solutions. In addition, with the aid of the Miura transformation, we also give the corresponding binary travelling-wave periodic solutions of KdV equation.     

Mathematics of dispersive water waves

A commuting hierarchy of dispersive water wave equations makes a three-Hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed. The modified water wave hierarchy is a bi-Hamiltonian system; its modification bifurcates. The water wave hierarchy, and the hierarchies of the Korteweg-de Vries and the modified Korteweg-de Vries equations, as well as the classical Miura map, are given new representations through various specializations of nonstandard systems.     

Fluctuating solitons of the KdV equations

We introduce a new AKNS three-component system, which is convenient for finding periodic and/or almost periodic solutions to the hierarchy of the KdV equations. It conserves the spectral functions which determine the spectrum of the auxiliary Schrödinger equation containing the solutions of the Korteweg-de Vries equations as potentials. By means of the Darboux and Abraham-Moses transformations we derive new solutions of the KdV hierarchy, which can be grasped as solitons on the fluctuating background.Some parts of the paper were delivered in the talk at the III Potsdam-V Kiev international workshop on nonlinear processes in physics, Potsdam (USA), 1–11 August, 1991.     

Some quasi-periodic solutions to the Kadometsev-Petviashvili and modified Kadometsev-Petviashvili equations

The Kadometsev-Petviashvili (KP) and modified KP (mKP) equations are retrieved from the first two soliton equations of coupled Korteweg-de Vries (cKdV) hierarchy. Based on the nonlinearization of Lax pairs, the KP and mKP equations are ultimately reduced to integrable finite-dimensional Hamiltonian systems in view of the r-matrix theory. Finally, the resulting Hamiltonian flows are linearized in Abel-Jacobi coordinates, such that some specially explicit quasi-periodic solutions to the KP and mKP equations are synchronously given in terms of theta functions through the Jacobi inversion.     

Linear superposition in nonlinear equations

Several nonlinear systems such as the Korteweg-de Vries (KdV) and modified KdV equations and lambda phi(4) theory possess periodic traveling wave solutions involving Jacobi elliptic functions. We show that suitable linear combinations of these known periodic solutions yield many additional solutions with different periods and velocities. This linear superposition procedure works by virtue of some remarkable new identities involving elliptic functions.     

Complex solitary waves and soliton trains in KdV and mKdV equations

We demonstrate the existence of complex solitary wave and periodic solutions of theKorteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations. The solutions ofthe KdV (mKdV) equation appear in complex-conjugate pairs and are even (odd) under thesimultaneous actions of parity (??) and time-reversal (??) operations. The corresponding localized solitons arehydrodynamic analogs of Bloch soliton in magnetic system, with asymptotically vanishingintensity. The ????-odd complex soliton solution is shown to beiso-spectrally connected to the fundamental sech² solution through supersymmetry. Physically, thesecomplex solutions are analogous to the experimentally observed grey solitons of non-liner Schödinger equation, governing the dynamics of shallow waterwaves and hence may also find physical verification.     

On the nonlinear Schrödinger limit of the Korteweg-de Vries equation

Using the multiscale approach of Zakharov and Kuznetsov it is shown that the nonlinear Schrödinger periodic scattering data is related to the Korteweg-de Vries periodic scattering data via an average over the Korteweg-de Vries carrier oscillation. This allows a complete elucidation of the physical meaning of the nonlinear Schrödinger scattering data, conservation laws, theta function solutions and reality constraint.     

Deriving the New Traveling Wave Solutions for the Nonlinear Dispersive Equation,KdV-ZK Equation and Complex Coupled KdV System Using Extended Simplest Equation Method

In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov-Kuznetsov (KdV-ZK) equation and complex coupled KdV system by using extended simplest equation method, and then derive the hyperbolic function solutions include soliton solutions, trigonometric function solutions include periodic solutions with special values for double parameters and rational solutions. The properties of such solutions are shown by figures. The results show that this method is an effective and a powerful tool for handling the solutions of nonlinear partial differential equations (NLEEs) in mathematical physics.     

Breather solutions of a negative order modified Korteweg-de Vries equation and its nonlinear stability

This paper is concerned with a negative order modified Korteweg-de Vries (nmKdV) equation which is in the negative flow of the mKdV hierarchy. We construct the breather solutions by Hirota's bilinear method and derive the infinite conservation laws through the Lax pair of the nmKdV equation. By constructing a new Lyapunov function with the conservation laws, we obtain the nonlinear stability of the breather solutions.     

Grassmannian and String Theory

The infinite-dimensional Grassmannian manifold contains moduli spaces of Riemann surfaces of all genera. This well known fact leads to a conjecture that non-perturbative string theory can be formulated in terms of the Grassmannian. We present new facts supporting this hypothesis. In particular, it is shown that Grassmannians can be considered as generalized moduli spaces; this statement permits us to define corresponding “string amplitudes” (at least formally). One can conjecture that it is possible to explain the relation between non-perturbative and perturbative string theory by means of localization theorems for equivariant cohomology; this conjecture is based on the characterization of moduli spaces, relevant to string theory, as sets consisting of points with large stabilizers in certain groups acting on the Grassmannian. We describe an involution on the Grassmannian that could be related to S-duality in string theory. Received: 19 December 1996 / Accepted: 27 March 1998