Schwarzian Korteweg-de Vries Research Articles

The (2+1)‐dimensional Schwarzian Korteweg–de Vries equation and its generalizations with discrete Lax matrices

By using the discrete Lax matrices corresponding to and in the Adler–Bobenko–Suris list of quadrilateral lattice equations, we establish solutions of the (2+1)‐dimensional Schwarzian Korteweg–de Vries (SKdV) equation and its generalizations. According to the structure of integrable Hamiltonian systems provided by one discrete Lax matrix for the lattice, we construct a novel Lax representation for the (2+1)‐dimensional SKdV equation. On the basis of the Riemann surface and elliptic variables, the Hamiltonian phase flows related to the equation are linearized by the Abel–Jacobi coordinates. Consequently, we obtain finite genus solutions to the (2+1)‐dimensional SKdV equation. From the discrete Lax matrix of the lattice, we study the generalized (2+1)‐dimensional SKdV equation with a parameter , which can be reduced to a simple case of the Krichever–Novikov equation. Finally, we show that the case of can also be solved through a modified version of the other discrete Lax matrix for the equation.

Computational simulations; propagation behavior of the Riemann wave interacting with the long wave

The analytical solutions to the (2+1)-dimensional integrable Schwarz-Korteweg-de Vries (SKdV) problem are investigated in this article. To get solitary wave formulae such as complex, exponential, hyperbolic, and trigonometric function solutions, the modified Khater technique and the improved Bernoulli sub-equation function method (IBSEFM) are used. Furthermore, their stability characteristics are investigated. The solutions produced using the two methods are compared in order to uncover relationships between them and their features as indicated by the appropriate parameter values.

Hamiltonian structures for integrable hierarchies of Lagrangian PDEs

Many integrable hierarchies of differential equations allow a variational description, called a Lagrangian multiform or a pluri-Lagrangian structure. The fundamental object in this theory is not a Lagrange function but a differential $d$-form that is integrated over arbitrary $d$-dimensional submanifolds. All such action integrals must be stationary for a field to be a solution to the pluri-Lagrangian problem. In this paper we present a procedure to obtain Hamiltonian structures from the pluri-Lagrangian formulation of an integrable hierarchy of PDEs. As a prelude, we review a similar procedure for integrable ODEs. We show that exterior derivative of the Lagrangian $d$-form is closely related to the Poisson brackets between the corresponding Hamilton functions. In the ODE (Lagrangian 1-form) case we discuss as examples the Toda hierarchy and the Kepler problem. As examples for the PDE (Lagrangian 2-form) case we present the potential and Schwarzian Korteweg-de Vries hierarchies, as well as the Boussinesq hierarchy.

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.

Computational and Numerical Solutions for 2+1-Dimensional Integrable Schwarz–Korteweg–de Vries Equation with Miura Transform

This paper investigates the analytical, semianalytical, and numerical solutions of the 2+1–dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation. The extended simplest equation method, the sech-tanh method, the Adomian decomposition method, and cubic spline scheme are employed to obtain distinct formulas of solitary waves that are employed to calculate the initial and boundary conditions. Consequently, the numerical solutions of this model can be investigated. Moreover, their stability properties are also analyzed. The solutions obtained by means of these techniques are compared to unravel relations between them and their characteristics illustrated under the suitable choice of the parameter values.

Darboux transformation and soliton solutions of the (2+1)-dimensional Schwarz–Korteweg–de Vries equation

In this paper, we deduce the (2+1)-dimensional Schwarz–Korteweg–de Vries equation from two (1+1)-dimensional equations. Based on the resulting Lax pairs, we present its [Formula: see text]-fold Darboux transformation. From a trivial solution, we get [Formula: see text]-soliton and [Formula: see text]-soliton solutions of the (2+1)-dimensional Schwarz–Korteweg–de Vries equation. The asymptotic analyses of the [Formula: see text]-soliton, [Formula: see text]-soliton and [Formula: see text]-soliton solutions are presented theoretically and graphically.

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.

Schwarzian derivative and Ermakov equation on a time scale

We introduce the Schwarzian, simplified Schwarzian, and Schwarzian Korteweg-de Vries equations on a time scale that are invariant under the fractional linear transformations. As an application, we derive their solutions and establish the invariant disconjugacy condition for second order dynamic equations on a time scale. Furthermore, we consider the Ermakov dynamic equation and the Ermakov-Lewis adiabatic invariant on a time scale. We discuss specific examples of discrete, quantum, and continuous time scales to compare our equations with the well-known ones. We also derive the linearization of the Ermakov equation and the corresponding Painleve equation on a time scale.

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.

New traveling wave solutions to AKNS and SKdV equations

We analyze the traveling wave solutions of Ablowitz-Kaup-Newell-Segur (AKNS) and Schwarz-Korteweg-de Vries (SKdV) equations. As the solution method for differential equations we consider the improved tanh approach. This approach provides to transform the partial differential equation into the ordinary differential equation and then obtain the new families of exact solutions based on the solutions of the Riccati equation. The different values of the coefficients of the Riccati equation allow us to obtain new type of traveling wave solutions to AKNS and SKdV equations.

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.

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.

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.

Traveling-Wave Solutions of the Schwarz–Korteweg–de Vries Equation in 2+1 Dimensions and the Ablowitz–Kaup–Newell–Segur Equation Through Symmetry Reductions

One of the more interesting solutions of the (2+1)-dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation is the soliton solutions. We previously derived a complete group classification for the SKdV equation in 2+1 dimensions. Using classical Lie symmetries, we now consider traveling-wave reductions with a variable velocity depending on the form of an arbitrary function. The corresponding solutions of the (2+1)-dimensional equation involve up to three arbitrary smooth functions. Consequently, the solutions exhibit a rich variety of qualitative behaviors. In particular, we show the interaction of a Wadati soliton with a line soliton. Moreover, via a Miura transformation, the SKdV is closely related to the Ablowitz–Kaup–Newell–Segur (AKNS) equation in 2+1 dimensions. Using classical Lie symmetries, we consider traveling-wave reductions for the AKNS equation in 2+1 dimensions. It is interesting that neither of the (2+1)-dimensional integrable systems considered admit Virasoro-type subalgebras.

Classical Symmetry Reductions of the Schwarz–Korteweg–de Vries Equation in 2+1 Dimensions

Classical reductions of a (2+1)-dimensional integrable Schwarz–Korteweg–de Vries equation are classified. These reductions to systems of partial differential equations in 1+1 dimensions admit symmetries that lead to further reductions, i.e., to systems of ordinary differential equations. All these systems have been reduced to second-order ordinary differential equations. We present some particular solutions involving two arbitrary functions.

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.

The investigation into the Schwarz–Korteweg–de Vries equation and the Schwarz derivative in (2+1) dimensions

In this note, we shall introduce a new integrable equation and the Schwarz derivative in (2+1) dimensions. First we show the existence of the Lax pair for an equation which has the relation to the Schwarz–Korteweg–de Vries (SKdV) equation. Next we derive a new equation in (2+1) dimensions by using a well-known higher-dimensional manner to the Lax pair for the SKdV equation. The (2+1) dimensional Schwarz derivative is defined here. Finally we briefly discuss various results which we have obtained about the new equation.