Schwarzian Korteweg-de Vries Equation Research Articles

Dynamical Behaviour of the Light Pulses through the Optical Fiber: Two Nonlinear Atangana Conformable Fractional Evolution Equations

This study, using the extended simplest method of equation, examines the explicit movement solutions of both the Schwarzian Korteweg-de Vries (SKdV) and (2 + 1)-Ablowitz-Kaup-Newell-Segur (AKNS.) equation. These models show the movement of the waves in optical fiber mathematically. The SKdV equation explains the movement of the isolated waves in diverse fields and on the site in a small space microsection. Some solutions obtained have been developed to show the physical and dynamic behaviors of these solutions in the obtained wave.

Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation

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.

Oscillatory Solutions for Lattice Korteweg-de Vries-Type Equations

Abstract By imposing some shift relations on r which satisfies the Sylvester equation KM + MK = r t c , oscillatory solutions are presented for some lattice Korteweg-de Vries-type equations, including the lattice potential Korteweg de-Vires equation, lattice potential modified Korteweg de-Vires equation, and lattice Schwarzian Korteweg-de Vries equation. This is done through the generalised Cauchy matrix approach.

The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation KM + MK = r sT we introduce a scalar function S(i, j) = sT Kj (I + M)−1Kir which is defined as same as in discrete case. S(i, j) satisfy some recurrence relations which can be viewed as discrete equations and play indispensable roles in deriving continuous integrable equations. By imposing dispersion relations on r and s, we find the Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian Korteweg-de Vries equation and sine-Gordon equation can be expressed by some discrete equations of S(i, j) defined on certain points. Some special matrices are used to solve the Sylvester equation and prove symmetry property S(i, j) = S(j,i). The solution M provides t function by t = ∣I + M∣. We hope our results can not only unify the Cauchy matrix approach in both continuous and discrete cases, but also bring more links for integrable systems and variety of areas where the Sylvester equation appears frequently.

Twisted reductions of integrable lattice equations, and their Lax representations

It is well known that from two-dimensional lattice equations one can derive one-dimensional lattice equations by imposing periodicity in some direction. In this paper we generalize the periodicity condition by adding a symmetry transformation and apply this idea to autonomous and non-autonomous lattice equations. As results of this approach, we obtain new reductions of the discrete potential Korteweg–de Vries (KdV) equation, discrete modified KdV equation and the discrete Schwarzian KdV equation. We will also describe a direct method for obtaining Lax representations for the reduced equations.

Non-commutative q-Painlevé VI equation

By applying suitable centrality conditions to non-commutative non-isospectral lattice modified Gel'fand–Dikii type systems we obtain the corresponding non-autonomous equations. Then we derive the non-commutative q-discrete Painlevé VI equation with full range of parameters as the (2, 2) similarity reduction of the non-commutative, non-isospectral and non-autonomous lattice modified Korteweg–de Vries equation. We also comment on the fact that in making the analogous reduction starting from the Schwarzian Korteweg–de Vries equation no such ‘non-isospectral generalization’ is needed.

Discrete Painlevé equations and their Lax pairs as reductions of integrable lattice equations

We describe a method to obtain Lax pairs for periodic reductions of a rather general class of integrable non-autonomous lattice equations. The method is applied to obtain reductions of the non-autonomous discrete Korteweg–de Vries equation and non-autonomous discrete Schwarzian Korteweg–de Vries equation, which yield a discrete analogue of the fourth Painlevé equation, a q-analogue of the sixth Painlevé equation and the q-Painlevé equation with a symmetry group of affine Weyl type E(1)6.

Analytic investigation of the (2 + 1)-dimensional Schwarzian Korteweg–de Vries equation for traveling wave solutions

By means of the two distinct methods, the Exp-function method and the extended (G′/G)-expansion method, we successfully performed an analytic study on the (2+1)-dimensional Schwarzian Korteweg–de Vries equation. We exhibited its further closed form traveling wave solutions which reduce to solitary and periodic waves. New rational solutions are also revealed.

The Exp-function approach to the Schwarzian Korteweg–de Vries equation

By means of the Exp-function method and its generalization, we report further exact traveling wave solutions, in a concise form, to the Schwarzian Korteweg–de Vries equation which admits physical significance in applications. Not only solitary and periodic waves but also rational solutions are observed.

Almost periodic solutions of a [formula omitted]-dimensional Schwarzian Korteweg de Vries equation

We consider the evolution of the ( 2 + 1 ) -dimensional Schwarzian Korteweg de Vries equation { 8 ϖ 2 ϖ τ + 2 ϖ 2 ϖ ζ ζ η − 8 ϖ ϖ ζ ϖ ζ η − 2 ϖ ϖ ζ ζ ϖ η + 4 ϖ ζ 2 ϖ η − ϖ ϖ ζ ϖ ̃ η = 0 , ϖ 2 ϖ ̃ ζ = ϖ ζ 2 , with ϖ ⁄ = 0 in C , while the independent variable τ (“time”) is complex. We prove that there exist many almost periodic solutions.

New solutions of the Schwarzian Korteweg-de Vries equation in 2+1 dimensions based on weak symmetries

We consider the (2+1)-dimensional integrable Schwarzian Korteweg-de Vries equation. Using weak symmetries, we obtain a system of partial differential equations in 1+1 dimensions. Further reductions lead to second-order ordinary differential equations that provide new solutions expressible in terms of known functions. These solutions depend on two arbitrary functions and one arbitrary solution of the Riemann wave equation and cannot be obtained by classical or nonclassical symmetries. Some of the obtained solutions of the Schwarzian Korteweg-de Vries equation exhibit a wide variety of qualitative behaviors; traveling waves and soliton solutions are among the most interesting.

New classes of solutions for the Schwarzian Korteweg–de Vries equation in (2+1) dimensions**Partially supported by projects MTM2006-05031 from DGES and FMQ201 from Junta de Andalucía.

We have obtained new classes of solutions for the (2+1)-dimensional Schwarzian Korteweg–de Vries equation by considering several types of reductions of a system equivalent to this equation. The first analysis is done by studying the nonclassical reductions of the system. Further reductions are attained by means of other types of symmetry reductions or by ansatz-based reductions. Most of the new classes of solutions depend on Jacobian elliptic functions and solutions of a Riemann wave equation, including the cnoidal waves solutions. The new classes of solutions can display several types of coherent structures and can exhibit the overturning or intertwining phenomena, according to the suitable selection of the functions these solutions depend on.

On completely integrable geometric evolutions of curves of Lagrangian planes

In this paper we find an explicit moving frame along curves of Lagrangian planes invariant under the action of the symplectic group. We use the moving frame to find a family of independent and generating differential invariants. We then construct geometric Hamiltonian structures in the space of differential invariants and prove that, if we restrict them to a certain Poisson submanifold, they become a set of decoupled Korteweg–de Vries (KdV) first and second Hamiltonian structures. We find an evolution of curves of Lagrangian planes that induces a system of decoupled KdV equations on their differential invariants (we call it the Lagrangian Schwarzian KdV equation). We also show that a generalized Miura transformation takes this system to a modified matrix KdV equation. In the four-dimensional case we show that there are no unrestricted compatible geometric pairs.

Second-order recursion operators of third-order evolution equations with fourth-order integrating factors

We report the recursion operators for a class of symmetry integrable evolution equations of third order which admit a fourth-order integrating factor. Under some assumptions we obtain the complete list of equations, one of which is a special case of the Schwarzian Korteweg-de Vries equation.

Multiple solutions for the Schwarzian Korteweg–de Vries equation in (2+1) dimensions

Abstract In this paper we find new families of solutions for the (2 + 1)-dimensional integrable Schwarzian Korteweg–de Vries equation, that depend up to two arbitrary functions and a solution of a Riemann wave equation. Some of these solutions exhibit a rich dynamic, with a wide variety of qualitative behavior and structures that are exponentially localized. We have also found several families of overturning and intertwining solutions for the equation, that correspond to the nonconstant solutions of Riemann equations.

A modified Schwarzian Korteweg-de Vries equation in 2 + 1 dimensions with lots of isochronous solutions

A modified version of the integrable Schwarzian Korteweg de Vries equation in 2 + 1 dimensions is introduced, and it is pointed out that it possesses lots of isochronous solutions.

The Schwarzian Korteweg de Vries equation in (2   1) dimensions

In this paper, the complete Lie group classification of a (2 + 1)-dimensional integrable Schwarzian Korteweg–de Vries equation is obtained. The reduction to systems of partial differential equations in (1 + 1) dimension is derived from the optimal system of subalgebras. The invariance study of these systems leads to second-order ODEs. These ODEs provide several classes of solutions; all of them are expressible in terms of known functions, some of them are expressible in terms of the second and third Painleve transcendents. The corresponding solutions of the (2 + 1)-dimensional equation involve up to three arbitrary smooth functions. They even appear in the form ρ(z)f(x + (t)). Consequently, the solutions exhibit a rich variety of qualitative behaviour. Indeed, by making appropriate choices for the arbitrary functions, we are able to exhibit large families of solitary waves, coherent structures and different types of bound states.

Diffeomorphisms on S1, projective structures and integrable systems

AbstractIn this paper we consider a projective connection as defined by the nth-order Adler-Gelfand-Dikii (AGD) operator on the circle. It is well-known that the Korteweg-de Vries (KdV) equation is the archetypal example of a scalar Lax equation defined by a Lax pair of scalar nth-order differential (AGD) operators. In this paper we derive (formally) the KdV equation as an evolution equation of the AGD operator (at least for n ≤ 4) under the action of Vect(S1). The solutions of the AGD operator define an immersion R → RPn−1 in homogeneous coordinates. In this paper we derive the Schwarzian KdV equation as an evolution of the solution curve associated with Δ(n), for n ≤ 4.