Analytical approach for the fractional differential equations by using the extended tanh method

In this study, we consider analytical solutions of space–time fractional derivative foam drainage equation, the nonlinear Korteweg–de Vries equation with time and space-fractional derivatives and time-fractional reaction–diffusion equation by using the extended tanh method. The fractional derivatives are defined in the modified Riemann–Liouville context. As a result, various exact analytical solutions consisting of trigonometric function solutions, kink-shaped soliton solutions and new exact solitary wave solutions are obtained.     

Quasi-periodic wave solutions,soliton solutions,and integrability to a (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation

In this paper, a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko (gBK) equation is investigated, which can be used to describe the interaction of a Riemann wave propagating along y-axis and a long wave propagating along x-axis. The complete integrability of the gBK equation is systematically presented. By employing Bell’s polynomials, a lucid and systematic approach is proposed to systematically study its bilinear formalism, bilinear Bäcklund transformations, Lax pairs, respectively. Furthermore, based on multidimensional Riemann theta functions, the periodic wave solutions and soliton solutions of the gBK equation are derived. Finally, an asymptotic relation between the periodic wave solutions and soliton solutions are strictly established under a certain limit condition.     

Invariant resolutions for several Fueter operators

A proper generalization of complex function theory to higher dimension is Clifford analysis and an analogue of holomorphic functions of several complex variables were recently described as the space of solutions of several Dirac equations. The four-dimensional case has special features and is closely connected to functions of quaternionic variables. In this paper we present an approach to the Dolbeault sequence for several quaternionic variables based on symmetries and representation theory. In particular we prove that the resolution of the Cauchy–Fueter system obtained algebraically, via Gröbner bases techniques, is equivalent to the one obtained by R.J. Baston (J. Geom. Phys. 1992).     

Quasi-Periodic Solutions and Asymptotic Properties for the Isospectral BKP Equation

In this paper, based on a Riemann theta function and Hirota's bilinear form, a straightforward way is presented to explicitly construct Riemann theta functions periodic waves solutions of the isospectral BKP equation. Once the bilinear form of an equation obtained, its periodic wave solutions can be directly obtained by means of an unified theta function formula and the way of obtaining the bilinear form is given in this paper. Based on this, the Riemann theta function periodic wave solutions and soliton solutions are presented. The relations between the periodic wave solutions and soliton solutions are strictly established and asymptotic behaviors of the Riemann theta function periodic waves are analyzed by a limiting procedure. The N-soliton solutions of isospectral BKP equation are presented with its detailed proof.     

Quasi-Periodic Solutions and Asymptotic Properties for the Isospectral BKP Equation

In this paper, based on a Riemann theta function and Hirota's bilinear form, a straightforward way is presented to explicitly construct Riemann theta functions periodic waves solutions of the isospectral BKP equation.Once the bilinear form of an equation obtained, its periodic wave solutions can be directly obtained by means of an unified theta function formula and the way of obtaining the bilinear form is given in this paper. Based on this, the Riemann theta function periodic wave solutions and soliton solutions are presented. The relations between the periodic wave solutions and soliton solutions are strictly established and asymptotic behaviors of the Riemann theta function periodic waves are analyzed by a limiting procedure. The N-soliton solutions of isospectral BKP equation are presented with its detailed proof.     

A discrete negative AKNS equation: generalized Cauchy matrix approach

Generalized Cauchy matrix approach is used to investigate a discrete negative Ablowitz–Kaup–Newell–Segur (AKNS) equation. Several kinds of solutions more than multi-soliton solutions to this equation are derived by solving determining equation set. Furthermore, applying an appropriate continuum limit we obtain a semidiscrete negative AKNS equation and after a second continuum limit we derive the nonlinear negative AKNS equation. The reductions to discrete, semi-discrete and continuous sine-Gordon equations are also discussed.     

修正的Korteweg de Vries-正弦Gordon方程的Riemannθ函数解

给出椭圆方程的一组Riemannθ函数解,结合它的一个Backlund变换,得到这个椭圆方程的无穷序列Riemannθ函数解.最后利用这个椭圆方程作为辅助方程,借助于符号计算软件Mathematica,得到了修正的Korteweg deVries-正弦Gordon方程的无穷序列Riemannθ函数解.     

On quasi-periodic solutions of the 2+1 dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation

The 2+1 dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation is decomposed into systems of integrable ordinary differential equations resorting to the nonlinearization of Lax pairs. The Abel–Jacobi coordinates are introduced to straighten the flows, from which quasi-periodic solutions of the 2+1 dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation are obtained in terms of Riemann theta functions.     

Solitons and periodic solutions to a couple of fractional nonlinear evolution equations

This paper studies a couple of fractional nonlinear evolution equations using first integral method. These evolution equations are foam drainage equation and Klein–Gordon equation (KGE), the latter of which is considered in (2 + 1) dimensions. For the fractional evolution, the Jumarie’s modified Riemann–Liouville derivative is considered. Exact solutions to these equations are obtained.     

A novel method for nonlinear fractional differential equations using symbolic computation

In this paper, a new approach, namely an ansatz method is applied to find exact solutions for nonlinear fractional differential equations in the sense of modified Riemann–Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to solve the fractional-order biological population model and the space–time fractional modified equal width equation, and as a result, some dark soliton solutions for them are established.     

非线性发展方程的Riemann theta 函数等几种新解

为了构造非线性发展方程的复合型无穷序列精确解, 获得了第二种椭圆方程的Riemann theta 函数等几种新解.在此基础上,利用第二种椭圆方程与Riccati方程的Bäcklund变换和解的非线性叠加公式, 借助符号计算系统 Mathematica, 以mKdV方程为应用实例, 构造了该方程的复合型无穷序列新精确解.这里包括Riemann theta 函数、Jacobi椭圆函数、双曲函数、 三角函数和有理函数,通过几种形式构成的复合型无穷序列新精确解. 关键词： 第二种椭圆方程 Riccati方程 非线性发展方程 Riemann theta 函数无穷序列解     

Exponential rational function method for space–time fractional differential equations

In this paper, exponential rational function method is applied to obtain analytical solutions of the space–time fractional Fokas equation, the space–time fractional Zakharov Kuznetsov Benjamin Bona Mahony, and the space–time fractional coupled Burgers’ equations. As a result, some exact solutions for them are successfully established. These solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie’s modified Riemann–Liouville sense. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.     

The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations

In this paper, we implemented the functional variable method and the modified Riemann–Liouville derivative for the exact solitary wave solutions and periodic wave solutions of the time-fractional Klein–Gordon equation, and the time-fractional Hirota–Satsuma coupled KdV system. This method is extremely simple but effective for handling nonlinear time-fractional differential equations.     

The Landau-Lifschitz equation and the Riemann boundary problem on a torus

The Cauchy problem for the Landau-Lifschitz equation is studied by the inverse transform method. We reduce this problem to the matrix Riemann problem on a torus and then to a certain Fredholm integral equation. An exact N-soliton solution is constructed. The reduction group approach is a main tool of our study.     

Discrete Riemann Surfaces and the Ising Model

We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy–Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality. Received: 23 May 2000/ Accepted: 21 November 2000     

Multi-cut solutions of Laplacian growth

A new class of solutions to Laplacian growth (LG) with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases. These solutions, which are time-dependent conformal maps with branch cuts inside the unit circle, are governed by a nonlinear integral equation and describe oil fjords with non-parallel walls in viscous fingering experiments in Hele-Shaw cells. Integrals of motion for the multi-cut LG solutions in terms of singularities of the Schwarz function are found, and the dynamics of densities (jumps) on the cuts are derived. The subclass of these solutions with linear Cauchy densities on the cuts of the Schwarz function is of particular interest, because in this case the integral equation for the conformal map becomes linear. These solutions can also be of physical importance by representing oil/air interfaces, which form oil fjords with a constant opening angle, in accordance with recent experiments in a Hele-shaw cell.     

Local index theorem for orbifold Riemann surfaces

We derive a local index theorem in Quillen’s form for families of Cauchy–Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated Fuchsian groups. Each conical point (or a conjugacy class of primitive elliptic elements in the Fuchsian group) gives rise to an extra term in the local index theorem that is proportional to the symplectic form of a new Kähler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local Kähler potential of the elliptic metric and show that when the order of elliptic element becomes large, the elliptic metric converges to the cuspidal one corresponding to a puncture on the orbisurface (or a conjugacy class of primitive parabolic elements). We also give a simple example of a relation between the elliptic metric and special values of Selberg’s zeta function.

     

Exactly solving some typical Riemann-Liouville fractional models by a general method of separation of variables

Finding exact solutions for Riemann–Liouville(RL) fractional equations is very difficult. We propose a general method of separation of variables to study the problem. We obtain several general results and, as applications, we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation. In particular, we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation. In addition, we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions.     

The short pulse equation by a Riemann–Hilbert approach

We develop a Riemann–Hilbert approach to the inverse scattering transform method for the short pulse (SP) equation

$$\begin{aligned} u_{xt}=u+\tfrac{1}{6}(u^3)_{xx} \end{aligned}$$

with zero boundary conditions (as \(|x|\rightarrow \infty \)). This approach is directly applied to a Lax pair for the SP equation. It allows us to give a parametric representation of the solution to the Cauchy problem. This representation is then used for studying the longtime behavior of the solution as well as for retrieving the soliton solutions. Finally, the analysis of the longtime behavior allows us to formulate, in spectral terms, a sufficient condition for the wave breaking.

     

A LARGE TIME ASYMPTOTICS FOR TRANSPARENT POTENTIALS FOR THE NOVIKOV–VESELOV EQUATION AT POSITIVE ENERGY

In the present paper we begin studies on the large time asymptotic behavior for solutions of the Cauchy problem for the Novikov–Veselov equation (an analog of KdV in 2 + 1 dimensions) at positive energy. In addition, we are focused on a family of reflectionless (transparent) potentials parameterized by a function of two variables. In particular, we show that there are no isolated soliton type waves in the large time asymptotics for these solutions in contrast with well-known large time asymptotics for solutions of the KdV equation with reflectionless initial data.