Variable Separation for (1+1)-Dimensional Nonlinear Evolution Equations with Mixed Partial Derivatives

We present basic theory of variable separation for (1+1)-dimensional nonlinear evolution equations with mixed partial derivatives. As an application, we classify equations u_xt=A(u,u_x)u_xxx+B(u,u_x) that admits derivative-dependent functional separable solutions (DDFSSs) and illustrate how to construct those DDFSSs with some examples.     

Functional Separable Solutions to Nonlinear Diffusion Equations by Group Foliation Method

We consider the functional separation of variables to the nonlinear diffusion equation with source and convection termut = (A(x)D(u)ux)x B(x)Q(u),Ax ≠ 0.The functional separation of variables to this equation is studied by using the group foliation method.A classification is carried out for the equations which admit the function separable solutions.As a consequence,some solutions to the resulting equations are obtained.     

Conditional Lie-Bäcklund Symmetry and New Variable Separation Solutions of the Third Order KdV-Type Equations

The functionally generalized variable separation solutions of a general KdV-type equations u_t=u_xxx + A(u, u_x)u_xx + B(u, u_x) are investigated by developing the conditional Lie-Bäcklund symmetry method. A complete classification of the considered equations, which admit multi-dimensional invariant subspaces governed by higher-order conditional Lie-Bäcklund symmetries, is presented. As a result, several concrete examples are provided to construct functionally generalized variable separation solutions of some resulting equations.     

Invariant Sets and Exact Solutions to Higher-Dimensional Wave Equations

The invariant sets and exact solutions of the （1 ＋ 2）-dimensional wave equations are discussed. It is shown that there exist a class of solutions to the equations which belong to the invariant set E0 = {u ： ux = vxF（u）,uy = vyF（u） }. This approach is also developed to solve （1 ＋ N）-dimensional wave equations.     

Invariant Sets and Exact Solutions to Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

In this paper, we introduce a new invariant set ˜E₀={u:u_x=f^ˊ(x)F(u)+ε [g^ˊ(x) -f^ˊ(x)g(x)]F(u)exp(-∫^u(1/F(z))dz), where f and g are some smooth functions of x, ε is a constant, and F is a smooth function to be determined. The invariant sets and exact solutions to nonlinear diffusion equation u_t=(D(u)u_x)_x+Q(x,u)u_x+P(x,u), are discussed. It is shown that there exist several classes of solutions to the equation that belong to the invariant set ˜E₀.     

Abundant New Explicit Exact Soliton—Like Solutions and Painleve Test for the Generalized Burgers Equation in （2＋1）—Dimensional Space

We obtain Backlund transformation and some new kink-like solitary wave solutions for the generalized Burgers equation in (2 1)-dimensional space,ut 1/2(uδy^-1ux)x-uxx=0,by using the extended homogeneous balance method.As is well known,the introduction of the concept of dromions (the exponentially localized solutions in (2 1)-dimensional space)has triggered renewed interest in (2 1)-dimensional soliton systems.The solutions obtained are used to show that the variable ux admits exponentially localized solutions rather than the physical field u(x,y,t) itself.In addition,it is shown that the equation passes Painleve test.     

Symmetries and Infinite Dimensional Lie Algebra of the Variable Coefficient KDV Equation

With the help of the conditional strong symmetry, we obtain the K symmetries, r symmetries, infinite dimensional Lie algebra and the corresponding infinite conservation quantities of the variable coefficient KdV equation u_t = h₁(t)(u_xxx +6uu_x) +4h₂(t)u_x - h₀(t)(2u+ xu_x) with h₀, h₁ and h₂ being three arbitrary functions of t.     

Solution of ODE u＂ ＋ p（u）（u＇）^2＋ q（u） = 0 and Applications to Classifications of All Single Travelling Wave Solutions to Some Nonlinear Mathematical Physics Equations

Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc., are reduced to an integrable ODE expressed by u＂ ＋p（u）（u＇）^2 ＋ q（u） = 0 whose generai solution can be given. Furthermore, combining complete discrimination system for polynomiai, the classifications of all single travelling wave solutions to these equations are obtained. The equation u＂＋p（u）（u＇）^2＋q（u） = 0 includes the equation （u＇）^2 = f（u） as a special case, so the proposed method can be also applied to a large number of nonlinear equations. These complete results cannot be obtained by any indirect method.     

Approximate derivative-dependent functional variable separation for quasi-linear diffusion equations with a weak source

By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source ut = （A（u）Ux）x ＋ eB（u, Ux）. A complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is listed. As a consequence, some approxi- mate solutions to the resulting perturbed equations are constructed via examples.     

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.     

New variable separation solutions for the generalized nonlinear diffusion equations

The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A(u, ux)uxx +B(u, ux) is studied by using the conditional Lie–Ba¨cklund symmetry method. The variant forms of the considered equations,which admit the corresponding conditional Lie–Ba¨cklund symmetries, are characterized. To construct functionally generalized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided.     

A New(γ_A,σ_B)-Matrix KP Hierarchy and Its Solutions

A new(γA,σB)-matrix KP hierarchy with two time series γA and σB,which consists of γA-flow,σB-flow and mixed γA and σB-evolution equations of eigenfunctions,is proposed.The reduction and constrained flows of(γA,σB)matrix KP hierarchy are studied.The dressing method is generalized to the(γA,σB)-matrix KP hierarchy and some solutions are presented.     

On the similarity solution of evolution equation u t =H(x,t, u,u x,u xx , ...)

Let $$\begin{gathered} u^* = u + \in \eta (x,{\text{ }}t,{\text{ }}u), \hfill \\ \hfill \\ \hfill \\ x^* = x + \in \xi (x, t, u{\text{),}} \hfill \\ \hfill \\ \hfill \\ {\text{t}}^{\text{*}} = {\text{ }}t + \in \tau {\text{(}}x,{\text{ }}t,{\text{ }}u), \hfill \\ \end{gathered}$$ be an infinitesimal invariant transformation of the evolution equation u _t =H(x,t,u,?u/?x,...,? ⁿ :u/?x ⁿ . In this paper we give an explicit expression for \(\eta ^{X^i }\) in the ‘determining equation’ $$\eta ^T = \sum\limits_{i = 1}^n {{\text{ }}\eta ^{X^i } {\text{ }}\frac{{\partial H}}{{\partial u_i }} + \eta \frac{{\partial H}}{{\partial u_{} }} + \xi \frac{{\partial H}}{{\partial x}} + \tau } \frac{{\partial H}}{{\partial t}},$$ where u _i =? ⁱ u/?x ⁱ . By using this expression we derive a set of equations with η, ξ, τ as unknown functions and discuss in detail the cases of heat and KdV equations.     

Extension of Variable Separable Solutions for Nonlinear Evolution Equations

We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separable solution. The new definitions can unify various kinds of variable separable solutions appearing in references. As application, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.     

带源项的变系数非线性反应扩散方程的精确解

本文利用广义条件对称方法对带源项的变系数非线性反应扩散方程 f(x)u_t=(g(x)D(u)u_x)_x+h(x)P(u)u_x+q(x)Q(u)进行研究. 当扩散项D(u)取u^m (m≠-1,0,1)和e^u两种重要情形时, 对该方程进行对称约化,得到了具有广义泛函分离变量形式的精确解. 这些精确解包含了该方程对应常系数情况下的解. 关键词： 广义条件对称 精确解 非线性反应扩散方程     

Classification and Functional Separable Solutions to Extended Nonlinear Wave Equations

The generalized conditional symmetry approach is applied to study the variable separation of the extended wave equations. Complete classification of those equations admitting functional separable solutions is obtained and exact separable solutions to some of the resulting equations are constructed.     

Classification and Functional Separable Solutions to Extended Nonlinear Wave Equations

The generalized conditional symmetry approach is applied to study the variable separation of the extended wave equations. Complete classification of those equations admitting functional separable solutions is obtained and exact separable solutions to some of the resulting equations are constructed.     

Measurement of /V(ub)/ using inclusive B-->X(u)lnu decays with a novel X(u)-reconstruction method

We report the measurement of an inclusive partial branching fraction for charmless semileptonic B decay and the extraction of /V(ub)/. Candidates for B-->X(u)lnu are identified with a novel X(u) reconstruction method based on neutrino reconstruction via missing 4-momentum and a technique called "simulated annealing." Based on 86.9 fb(-1) of data taken with the Belle detector, we obtain DeltaB(B-->X(u)lnu;M(X)<1.7 GeV/c2,q2>8.0 GeV2/c2)=[7.37+/-0.89(stat)+/-1.12(syst)+/-0.55(b-->c)+/-0.24(b-->u)]x10(-4) and determine |V(ub)|=[4.66+/-0.28(stat)+/-0.35(syst)+/-0.17(b-->c)+/-0.08(b-->u)+/-0.58(theory)]x10(-3).     

New Similarity Reductions and Compacton Solutions for Boussinesq-Like Equations with Fully Nonlinear Dispersion

In this paper, similarity reductions of Boussinesq-like equations with nonlinear dispersion (simply called B(m,n) equations) u_tt=(uⁿ)_xx+(u^m)_xxxx, which is a generalized model of Boussinesq equation u_tt=(u²)_xx+u_xxxx and modified Bousinesq equation u_tt=(u³)_xx+u_xxxx, are considered by using the direct reduction method. As a result, several new types of similarity reductions are found. Based on the reduction equations and some simple transformations, we obtain the solitary wave solutions and compacton solutions (which are solitary waves with the property that after colliding with other compacton solutions, they re-emerge with the same coherent shape) of B(1,n) equations and B(m,m) equations, respectively.     

General Solution and Localized COherent Soliton Structures of the （2＋1）—Dimensional Generalized Davey—Stewarson Equations

In this paper,the variable separation approach is used to obtain localized coherent structures of the (2 1)-dimensional generalized Davey-Stewarson equations:iqt 1/2(qxx=qyy) (R S)q=0,Rx=-σ/2|q|y^2,Sy=-σ/2|q|2/x.Applying a special Baecklund transformation and introducing arbitrary functions of the seed solutions.and abundance of the localized structures of this model is derived,By selceting the arbitrary functions appropriately,some special types of localized excitations such as dromions,dromion lattice,breathers,and instantons are constructed.