A complex travelling wave solution to the KdV–Burgers equation

In the present work, making use of the hyperbolic tangent method a complex travelling wave solution to the KdV–Burgers equation is obtained. It is observed that the real part of the solution is the combination of a shock and a solitary wave whereas the imaginary part is the product of a shock with a solitary wave. By imposing some restrictions on the field variable at infinity, two complex waves, i.e., right going and left going waves with specific wave speed are obtained.     

Conformal invariance,the KdV equation and coadjoint orbits of the Virasoro algebra

We use the transformation properties of the energy-momentum tensor of 2-dimensional conformal field theories to study the symplectic geometry of DiffS¹/S¹ together with the other coadjoint orbits of the Virasoro algebra in the space of quadratic differentials. Also, we examine orbits of the N = 1 and N = 2 super-Virasoro algebras and investigate the connections with nonlinear differential equations of KdV type. The importance that the dual of the Virasoro algebra may have in the geometric formulation of string theory is discussed as well.     

Kac-Moody algebra,nonlocal symmetries,and backlund transformation for KdV equation

Gauge transformation of the Lax eigenfunction through the explicit use of Lie group generators is seen to generate a two-parameter Backlund transformation. Explicit integration of this in two particular cases leads to sech² type and rational solutions starting from the trivial one. A method is indicated to generate infinitesimal transformations aroundu in the sense of Steudel, which in this case leads to a nonlocal structure of transformations.     

A (2+1)-dimensional KdV equation and mKdV equation: Symmetries,group invariant solutions and conservation laws

In this paper, we analyse the (2+1)-dimensional KdV and mKdV equations. Firtly, on the basis of the extended Lax pair, we derive these equations. Thereafter, the symmetry generators are determined followed by the application of the mCK method. Finally, conservation laws (including higher order) are studied.     

改进的tanh函数方法与广义变系数KdV和MKdV方程新的精确解

利用改进的tanh函数方法将广义变系数KdV方程和MKdV方程化为一阶变系数非线性常微分方 程组-通过求解这个变系数非线性常微分方程组，获得了广义变系数KdV方程和MKdV方程新的 精确类孤子解、有理形式函数解和三角函数解- 关键词： 改进的tanh函数方法 类孤子解 有理形式函数解 三角函数解     

New explicit exact solutions for a generalized Hirota—Satsuma coupled KdV system and a coupled MKdV equation

In this paper, we make use of a new generalized ansatz in the homogeneous balance method, the well-known Riccati equation and the symbolic computation to study a generalized Hirota--Satsuma coupled KdV system and a coupled MKdV equation, respectively. As a result, numerous explicit exact solutions, comprising new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained.     

Continuous feedback control of KdV Burgers system

The chaos in the KdV Burgers equation describing a ferroelectric system has been successfully controlled by using a continuous feedback control. This system has two stationary points. In order to know whether the chaos is controlled or not, the instability of control equation has been analysed numerically. The numerical analysis shows that the chaos can be converted to one point by using one control signal, however, it can converted to the other point by using three control signals. The chaotic motion is converted to two desired stationary points and periodic orbits in numerical experiment separately.     

Simple Soliton Solution Method for the Combined KdV and MKdV Equation

Malfliet first proposed a simple solution method for the multisoliton solutionofthe KdV equation. Abdel-Rahman used Malfliet's method in a slightly modifiedform, and gave the multisoliton solution of the mKdV equation, RLW equation,Boussinesq equation, and modified Boussinesq equation. In this paper, we solvethe soliton solution of the cKdV=nmKdV equation by using this method.     

Pade approximants and closed form solutions of the KdV and MKdV equations

The closed form solutions of the Kortewegde Vries (KdV) and modified MKdV equations which are obtainable by inverse scattering or Hirota's method are found to be Pade approximants to the formal series which results from iterating particular solutions of the linearized equation.     

On invariant states and the commutant of a group of quasi-free automorphisms of the CAR algebra,II

We improve Theorem 3 of a previous paper [3] as follows: If M is a von Neumann algebra on the one-particle space which does not contain any finite type I factor direct summands, an automorphism of the CAR algebra which carries the set of quage-invariant quasi-free states Φ_A with A in M onto itself is quasi-free.     

The exact solutions to the complex KdV equation

In this Letter, Wronskian solutions for the complex KdV equation are obtained by Hirota's bilinear method. Moreover, starting from the bilinear Bäcklund transformation, multi-soliton solutions are presented for the same equation. At the same time, it is also shown that these two kinds of solutions are equivalent.     

On the exponentiation of the affine metaplectic lie algebra

This paper concerns the infinite polynomials with maximal degree 2 of creation and annihilation operators, which give a Fock space representation of the complexification of the affine symplectic group. We study exponentiability of these operators, and obtain explicitly the local connection between the complexification of the affine metaplectic representation and the corresponding Lie algebra.     

Space time fractional KdV Burgers equation for dust acoustic shock waves in dusty plasma with non-thermal ions

The KdV–Burgers equation for dust acoustic waves in unmagnetized plasma having electrons, singly charged nonthermal ions, and hot and cold dust species is derived using the reductive perturbation method. The Boltzmann distribution is used for electrons in the presence of the cold(hot) dust viscosity coefficients. The semi-inverse method and Agrawal variational technique are applied to formulate the space–time fractional KdV–Burgers equation which is solved using the fractional sub-equation method. The effect of the fractional parameter on the behavior of the dust acoustic shock waves in the dusty plasma is investigated.     

Boson expansion of lie algebras: The fermion pair algebra

A unitary boson expansion theory for Lie algebras with ladder representations is discussed and applied to the fermion pair algebra SO(2n).     

Integrable discretizations of the KdV equation

Several dynamical systems are constructed having Lax representations with a spectral parameter and turning into the KdV equation in a continuous limit. A new hamiltonian integro-differential equation is derived having a Lax representation and an infinite set of conserved quantities. Integrable Lie algebraic generalizations of the Volterra system are pointed out.     

Solitons of the modified KdV equation

We determine all reflectionless potentials for the one-dimensional charge symmetric Dirac operator, identify them as solitons of the modified KdV equation, and give the connection to the KdV solitons. An associated dynamical system is shown to be completely integrable.     

The extended Fan's sub-equation method and its application to KdV–MKdV,BKK and variant Boussinesq equations

An extended Fan's sub-equation method is used for constructing exact travelling wave solutions of nonlinear partial differential equations (NLPDEs). The key idea of this method is to take full advantage of the general elliptic equation involving five parameters which has more new solutions and whose degeneracies can lead to special sub-equations involving three parameters. More new solutions are obtained for KdV–MKdV, Broer–Kaup–Kupershmidt (BKK) and variant Boussinesq equations. Then we present a technique which not only gives us a clear relation among this general elliptic equation and other sub-equations involving three parameters (Riccati equation, first kind elliptic equation, auxiliary ordinary equation, generalized Riccati equation and so on), but also provides an approach to construct new exact solutions to NLPDEs.     

On invariant states and the commutant of a group of quasi-free automorphisms of the car algebra

We study primary states of the CAR algebra which are left invariant under quasi-free automorphisms α_U corresponding to unitaries U of a von Neumann algebra $\text{M}$ on the one-particle Hilbert space, and show that they are quasi-free states ?_A corresponding to self-adjoint operators A in $\text{M}$′ with 0 ? A ? 1, under the assumption that $\text{M}$ does not contain any finite type Ifactor direct summands. Next we study automorphisms of the CAR algebra which commute with α_U for U in a von Neumann algebra $\text{M}$ and show that they are quasi-free automorphisms α_U with U in $\text{M}$′ under the same assumption on $\text{M}$ as above. Finally by using the latter result we obtain a generalization of a theorem of Hugenholtz and Kadison [3].