In this paper, we consider the complex modified Korteweg-de Vries (mKdV) equation as a model of few-cycle optical pulses. Using the Lax pair, we construct a generalized Darboux transformation and systematically generate the first-, second- and third-order rogue wave solutions and analyze the nature of evolution of higher-order rogue waves in detail. Based on detailed numerical and analytical investigations, we classify the higher-order rogue waves with respect to their intrinsic structure, namely, fundamental pattern, triangular pattern, and ring pattern. We also present several new patterns of the rogue wave according to the standard and non-standard decomposition. The results of this paper explain the generalization of higher-order rogue waves in terms of rational solutions. We apply the contour line method to obtain the analytical formulas of the length and width of the first-order RW of the complex mKdV and the NLS equations. In nonlinear optics, the higher-order rogue wave solutions found here will be very useful to generate high-power few-cycle optical pulses which will be applicable in the area of ultra-short pulse technology.
Hirota's discrete Korteweg-de Vries equation (dKdV) is an integrable partial difference equation on 2-dimensional integer lattice, which approaches the Korteweg-de Vries equation in a continuum limit. We find new transformations to other equations, including a second-degree second-order partial difference equation, which provide an unusual embedding into a three-dimensional lattice. The consistency of the resulting system extends a property that has been widely used to study partial difference equations on multidimensional lattices.
We perform enhanced Lie symmetry analysis of generalized fifth-order Korteweg-de Vries equations with time-dependent coefficients. The corresponding similarity reductions are classified and some exact solutions are constructed.
This is a continuation of Ref.[1](arXiv:nlin.SI/0603008). In the present paper we review solutions to the modified Korteweg-de Vries equation in terms of Wronskians. The Wronskian entry vector needs to satisfy a matrix differential equation set which contains complex operation. This is different from the case of the Korteweg-de Vries equation. We introduce an auxiliary matrix to deal with the complex operation and then we are able to give complete solution expressions for the matrix differential equation set. The obtained solutions to the modified Korteweg-de Vries equation can simply be categ…
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction to this method.
We solve the Cauchy problem for the Korteweg-de Vries equation with initial conditions which are steplike Schwartz-type perturbations of finite-gap potentials under the assumption that the respective spectral bands either coincide or are disjoint.
In this paper we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg-de Vries-type (KdV-type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symple…
A differential-algebraic approach to studying the Lax type integrability of the generalized Riemann type hydrodynamic equations at N = 3; 4 is devised. The approach is also applied to studying the Lax type integrability of the well known Korteweg-de Vries dynamical system.
We study numerically the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+ε^{2}u_{xxx}=0$ for $ε\ll1$ and give a quantitative comparison of the numerical solution with various asymptotic formulae for small $ε$ in the whole $(x,t)$-plane. The matching of the asymptotic solutions is studied numerically.
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation with steplike initial data leading to a rarefaction wave. In addition to the leading asymptotic we also compute the next term in the asymptotic expansion of the rarefaction wave, which was not known before.
In the paper two kinds of solutions are derived for the complex Korteweg-de Vries equation, including 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 Korteweg-de Vries equation. Using the transformation, solitons, breathers and rational solutions to the complex Korteweg-de Vries equation are obtained from those of the modified Korteweg-de Vries equation. Dynamics of the obtained solutions are illustrated.
In this paper, we consider three semi-discrete modified Korteweg-de Vries type equations which are the nonlinear lumped self-dual network equation,the semi-discrete lattice potential modified Korteweg-de Vries equation and a semi-discrete modified Korteweg-de Vries equation. We derive several kinds of exact solutions, in particular rational solutions, in terms of the Casorati determinant for these three equations respectively. For some rational solutions, we present the related asymptotic analysis to understand their dynamics better.
We study the existence and stability of periodic traveling-wave solutions for complex modified Korteweg-de Vries equation. We also discuss the problem of uniform continuity of the data-solution mapping.
With the assistance of one fold Darboux transformation formula, we derive rogue wave solutions of the complex modified Korteweg-de Vries equation on an elliptic function background. We employ an algebraic method to find the necessary squared eigenfunctions and eigenvalues. To begin we construct the elliptic function background. Then, on top of this background, we create a rogue wave. We demonstrate the outcome for three distinct elliptic modulus values. We find that when we increase the modulus value the amplitude of rogue waves on the dn-periodic background decreases whereas it increases in t…
In this paper, we take advantage of the bilinearization reduction method to consider the local and nonlocal reduction of a discrete Ablowitz-Kaup-Newell-Segur equation. Exact solutions in double Casoratian form to the reduced nonlocal discrete modified Korteweg de-Vries equations are constructed. The dynamics of soliton solutions are analyzed and illustrated by asymptotic analysis. Moreover, both semi-continuous limit and full continuous limit, are applied to obtain the local and nonlocal semi-discrete modified Korteweg de-Vries equations, as well as the local and nonlocal continuous modified …
Two page encyclopedic article about the Korteweg-de Vries equation covering historical perspective, solitary wave and periodic solutions, modern developments, properties and applications, and further reading.
We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson formalism (e.g., for equations of Korteweg-de Vries type), geometry of Liouville-type hyperbolic systems (including the 2D Toda chains), and Euler-Lagrange gauge theories (such as the Yang-Mills theories, gravity, or the Poisson sigma-models). Also, we formulate several open problems.
We consider the vector generalization of the modified Korteweg-de Vries equation. We develop the inverse scattering transform for solving this equation. We construct the solitons and the breather solutions and investigate the processes of their collisions. We show that along with one-component soliton solutions, there are solutions which have essentially three-component structure.
We solve the Gardner deformation problem for the N=2 supersymmetric a=4 Korteweg-de Vries equation (P. Mathieu, 1988). We show that a known zero-curvature representation for this superequation yields the system of new nonlocal variables such that their derivatives contain the Gardner deformation for the classical KdV equation.
Within the (2 + 1)-dimensional Korteweg-de Vries equation framework, new bilinear Backlund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation. By introducing an arbitrary function, a family of deformed soliton and deformed breather solutions are presented with the improved Hirotas bilinear method. Choosing the appropriate parameters, their interesting dynamic behaviors are shown in three-dimensional plots. Furthermore, novel rational solutions are generated by taking the limit of obtained solitons. Additionally, two dimensional [2D] rogue wa…
Based on integrable Hamiltonian systems related to the derivative Schwarzian Korteweg-de Vries (SKdV) equation, a novel discrete Lax pair for the lattice SKdV (lSKdV) equation is given by two copies of a Darboux transformation which can be used to derive an integrable symplectic correspondence. Resorting to the discrete version of Liouville-Arnold theorem, finite genus solutions to the lSKdV equation are calculated through Riemann surface method.
We solve the Cauchy problem for the Korteweg-de Vries equation with steplike quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of derivatives with finite moments.
The modified method of simplest equation is applied to the extended Korteweg - de Vries equation and to generalized Camassa - Holm equation. Exact traveling wave solutions of these two nonlinear partial differential equations are obtained. The equations of Bernoulli, Riccati and the extended tanh - equation are used as simplest equations. Some of the obtained solutions correspond to surface water waves.
We solve the Cauchy problem for the modified Korteweg--de Vries equation with steplike quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of derivatives and moments finite.
Consider a one-dimensional Schroedinger operator which is a short range perturbation of a finite-gap operator. We give necessary and sufficient conditions on the left, right reflection coefficient such that the difference of the potentials has finite support to the left, right, respectively. Moreover, we apply these results to show a unique continuation type result for solutions of the Korteweg-de Vries equation in this context. By virtue of the Miura transform an analogous result for the modified Korteweg-de Vries equation is also obtained.
The supercomplexification is a special method of N=2 supersymmetrization of the integrable equations in which the bosonic sector could be reduced to the complex version of these equations. The N=2 supercomplex Korteweg de Vries, Sawada-Kotera and Kaup-Kupershmidt equations are defined and investigated. The common attribute of the supercomplex equations is appearance of the odd hamiltonian structures and superfermionic conservation laws. The odd bi-hamiltonian structure, Lax representation and superfermionic conserved currents for new N=2 supersymmetric Korteweg de Vries equation and for Sawada…
We present a numerical approach for generalised Korteweg-de Vries (KdV) equations on the real line. In the spatial dimension we compactify the real line and apply a Chebyshev collocation method. The time integration is performed with an implicit Runge-Kutta method of fourth order. Several examples are discussed: initial data bounded but not vanishing at infinity as well as data not satisfying the Faddeev condition, i.e. with a slow decay towards infinity.
In this paper we show that all algebro-geometric finite gap solutions to the Korteweg--de Vries equation can be realized as a limit of N-soliton solutions as N diverges to infinity (see remark 1 for the precise meaning of this statement). This is done using the the primitive solution framework initiated by [5,28,31]. One implication of this result is that the N-soliton solutions can approximate any bounded periodic solution to the Korteweg--de Vries equation arbitrarily well in the limit as N diverges to infinity. We also study primitive solutions numerically that have the same spectral proper…
The Korteweg-de Vries and Benjamin-Ono nonlinear wave equations can describe solitary waves, all of which propagate in the same direction and which emerge from collisions with their shapes unchanged. There are technical challenges to giving a description of the thermal equilibrium of a dilute gas of such solitons. These challenges can be overcome by considering a stochastic source of new solitons located at $x=0$. The total intensity and the momentum distributions of the solitons emitted by such sources determine the equilibrium distribution of soliton momenta for $x>0$.
