NEW SYMMETRIES AND LIE ALGEBRAS OF THE GENERAL KdV EQUATION

The strong symmetry of the general KdV equation is factorized to a simple form and then the inverse strong symmetry is obtained explicitly. Acting a strong symmetry of the general KdV equation on the trivial symmetry and the known re symmetry, we obtain four new sets of symmetries of the general KdV equation. All these sets of symmetries constitute an infinite dimensional Lie algebra.     

变系数广义KdV方程新的类孤波解和精确解

用普通KdV方程作变换，构造变系数广义KdV方程的解，获得了变系数广义KdV方程新的Jacobi椭圆函数精确解、类孤波解、三角函数解和Weierstrass椭圆函数解. 关键词： KdV方程 变系数广义KdV方程 类孤波解 精确解     

(3+1)维 KdV 方程新的精确解和孤子解

用普通Korteweg-de Vries(KdV)方程作变换,构造(3 1)维KdV方程的解,获得了新的孤子解、Jaoobi椭圆函数解、三角函数解和Weierstrass椭圆函数解.     

A general mapping approach and new travelling wave solutions to the general variable coefficient KdV equation

A general mapping deformation method is applied to a generalized variable coefficient KdV equation. Many new types of exact solutions, including solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions and other exact excitations are obtained by the use of a simple algebraic transformation relation between the generalized variable coefficient KdV equation and a generalized cubic nonlinear Klein－Gordon equation.     

Deforming the Travelling Wave Solutions of the KdV Equation to Those of the Generalized Fifth Order KdV Equation

Using some limiting procedures, the solutions of the fifth order KdV equation u_t + (μu²+ υu_xx + αuu_xx + βu_x² + γu³ + δu_xxxx)_x = 0 would degenerate into the solutions of a simple equation, say KdV equation. In this letter, we analyze the possibility of the inverse procedure of the limiting process mentioned above for the travelling wave solutions. The results show that the procedure for deforming a travelling wave solution of the KdV equation to that of the generalized fifth order KdV equation can be accomplished by some pure algebraic tricks. Moreover, this inverse procedure is not unique in general.     

The theoretical analysis of the lattice hydrodynamic models for traffic flow theory

The lattice hydrodynamic model is not only a simplified version of the macroscopic hydrodynamic model, but also connected with the microscopic car following model closely. The modified Korteweg-de Vries (mKdV) equation related to the density wave in a congested traffic region has been derived near the critical point since Nagatani first proposed it. But the Korteweg-de Vries (KdV) equation near the neutral stability line has not been studied, which has been investigated in detail for the car following model. We devote ourselves to obtaining the KdV equation from the original lattice hydrodynamic models and the KdV soliton solution to describe the traffic jam. Especially, we obtain the general soliton solution of the KdV equation and the mKdV equation. We review several lattice hydrodynamic models, which were proposed recently. We compare the modified models and carry out some analysis. Numerical simulations are conducted to demonstrate the nonlinear analysis results.     

KdV方程与高阶KdV方程行波解之间的形变理论

从具有较多参数的复杂非线性方程的解出发,取某些参数的极限,这些解就退化到具有较少参数的简单方程的解,本文以KdV方程的行波解为例说明对于某些特解,上述极限过程的逆过程,即将简单方程的解形变到复杂方程的解是可能的,但一般来说并不一定具有唯一性。 关键词：     

New exact solutions to the generalized KdV equation with generalized evolution

In this paper, by using a transformation and an application of Fan subequation, we study a class of generalized Korteweg–de Vries (KdV) equation with generalized evolution. As a result, more types of exact solutions to the generalized KdV equation with generalized evolution are obtained, which include more general single-hump solitons, multihump solitons, kink solutions and Jacobian elliptic function solutions with double periods.     

A note on the wronskian form of solutions of the KdV equation

We show here that the KdV equation has solutions in the wronskian form under more general conditions than those considered previously by other authors. These conditions are used to generate new types of solutions of the KdV in the wronskian form.     

Gauge Transformation and Generalized Nonlocal Symmetries of the Korteweg-de Vries Equation

After extending the usual Lax pair of the Korteweg-de Vries (KdV) equation to a generalized form by using a gauge transformation, an adjoint Lax pair of the KdV equation is introduced. With the help of the spectral functions of the Lax pair and the adjoint Lax pair, a new nonlocal seed symmetry (which is gauge-invariant) is found and then a set of new infinitely many generalized nonlocal symmetries are obtained after establishing a general symmetry theory for an arbitrary nonlinear system.     

A Variable Separation Approach to Solve the Integrable and Nonintegrable Models:Coherent Structures of the (2 + 1)-Dimensional KdV Eqnation

We study the localized coherent structures ofa generally nonintegrable (2 1 )-dimensional KdV equation via a variable separation approach. In a special integrable case, the entrance of some arbitrary functions leads to abundant coherent structures. However, in the general nonintegrable case, an additional condition has to be introduced for these arbitrary functions. Although the additional condition has been introduced into the solutions of the nonintegrable KdV equation, there still exist many interesting solitary wave structures. Especially, the nonintegrable KdV equation possesses the breather-like localized excitations, and the similar static ring soliton solutions as in the integrable case. Furthermor,in the integrable case, the interaction between two travelling ring solitons is elastic, while in the nonintegrable case we cannot find even the single travelling ring soliton solution.     

A Note on the Transformation of Variables of KP Equation,Cylindrical KP Equation and Spherical KP Equation

In a recent article(Commun. Theor. Phys. 67(2017) 207), three(2+1)-dimensional equations — KP equation, cylindrical KP equation and spherical KP equation, have been reduced to the same Kd V equation by using different transformation of variables, respectively. In this short note, by adding an adjustment item to original transformation, three more general transformation of variables corresponding to above three equations have been given.Substituting the solutions of the Kd V equation into our transformation of variables, more new exact solutions of the three(2+1)-dimensional equations can be obtained.     

The almost periodic solutions to the equations of the KdV hierarchy

The Milne equation is used as the auxiliary equation instead of the usual Schrödinger equation, when the almost periodic solutions of the KdV hierarchy are looked for. Its almost periodic solution is found for the finite band (gap) potential and its time dependence is determined in the general case, provided the potential satisfies the KdV equations. Further investigated problems are: trace formulas and their applications, the Bloch functions, conserved quantities, the Poisson brackets and the Hamiltonian system.     

An alternative derivation of the pulse solutions of the KdV hierarchy

An alternative approach issues from the Appelle transformation of the Schrödinger equation. One solves the inverse problem for the transformed equation, a general solution of which is a quadratic form of two independent solutions of the primary Schrödinger equation. If the potential in the Schrödinger equation obeys one equation of the KdV hierarchy, the time derivative of this form is a linear combination of the form and its space derivative. The coefficients in the combination depend on the potential and the energy parameter of the Schrödinger equation only. This relation also determines the time dependence of the spectral data which along with the solution of the inverse problem gives the solution of the KdV equations as usual.     

Ion-acoustic waves in plasma of warm ions and isothermal electrons using time-fractional KdV equation

The ion-acoustic solitary wave in collisionless unmagnetized plasma consisting of warm ions-fluid and isothermal electrons is studied using the time fractional KdV equation. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation for small but finite amplitude ion-acoustic wave in warm plasma. The Lagrangian of the time fractional KdV equation is used in a similar form to the Lagrangian of the regular KdV equation with fractional derivative for the time differentiation. The variation of the functional of this Lagrangian leads to the Euler-Lagrange equation that gives the time fractional KdV equation. The variational-iteration method is used to solve the derived time fractional KdV equation. The calculations of the solution are carried out for different values of the time fractional order. These calculations show that the time fractional can be used to modulate the electrostatic potential wave instead of adding a higher order dissipation term to the KdV equation. The results of the present investigation may be applicable to some plasma environments,such as the ionosphere plasma.     

Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics

In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the K(m, n) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method is proposed.     

A new method of new exact solutions and solitary wave-like solutions for the generalized variable coefficients Kadomtsev——Petviashvili equation

Using the solution of general Korteweg--de Vries (KdV) equation, the solutions of the generalized variable coefficient Kadomtsev--Petviashvili (KP) equation are constructed, and then its new solitary wave-like solution and Jacobi elliptic function solution are obtained.     

New longitudinal waves in electron-positron-ion quantum plasmas

A general quantum dispersion equation for electron-positron(hole)-ion quantum plasmas is derived and studied for some interesting cases. In an electron-positron-ion degenerate Fermi gas, with or without the Madelung term, a new type of zero sound waves are found. Whereas in an electron-hole-ion plasmas a new longitudinal quantum waves are revealed, which have no analogies in quantum electron-ion plasmas. The excitation of these quantum waves by a low-density monoenergetic straight electron beam is examined. Furthermore, the Korteweg-de Vries (KdV) equation for novel quantum waves is derived and the contribution of the Madelung term in the formation of the KdV solitons is discussed.     

A Limit Symmetry of Modified KdV Equation and Its Applications

In this letter we consider a limit symmetry of the modified KdV equation and its application. The similarity reduction leads to limit solutions of the modified KdV equation. Besides, a modified KdV equation with new self-consistent sources is obtained and its solutions are derived.     

An extended functional transformation method and its application in some evolution equations

In this paper, an extended functional transformation is given to solve some nonlinear evolution equations. This function, in fact,is a solution of the famous KdV equation, so this transformation gives a transformation between KdV equation and other soliton equations. Then many new exact solutions can be given by virtue of the solutions of KdV equation.