Application of Exp-function method to potential Kadomtsev–Petviashvili equation

Exact periodic kink-wave solution, periodic soliton and doubly periodic solutions for the potential Kadomtsev–Petviashvii (PKP) equation are obtained using Exp-function method with the help of Maple computation.     

On a class of elliptic solutions of the Boussinesq equations

An ansatz is considered for Krichever curves associated with doubly periodic finite-gap solutions of the Boussinesq equation expressed in terms of equi-anharmonic elliptic Weierstrass &-functions. Some examples are given.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 3, pp. 347–356, December, 1996.     

Doubly periodic solutions of the modified Kawahara equation

Some doubly periodic (Jacobi elliptic function) solutions of the modified Kawahara equation are presented in closed form. Our approach is to introduce a new auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly periodic solutions of the modified Kawahara equation. When the module m → 1, these solutions degenerate to the exact solitary wave solutions of the equation. Then we reveal the relation of some exact solutions for the modified Kawahara equation obtained by other authors.     

Existence of doubly periodic vortices in a generalized Chern–Simons model

We establish an existence theorem for the doubly periodic vortices in a generalized self-dual Chern–Simons model. We show that there exists a critical value of the coupling parameter such that there exist self-dual doubly periodic vortex solutions for the generalized self-dual Chern–Simons equation if and only if the coupling parameter is less than or equal to the value. The energy, magnetic flux, and electric charge associated to the field configurations are all specifically quantized. By the solutions obtained for this generalized self-dual Chern–Simons equation we can also construct doubly periodic vortex solutions to a related generalized self-dual Abelian Higgs equation.     

On Doubly Periodic Standing Wave Solutions of the Coupled Higgs Field Equation

In this paper, we derive a class of doubly periodic standing wave solutions for a coupled Higgs field equation by employing the Hirota bilinear method and theta function identities. Such solutions are expressed in terms of theta functions with variable separation form. Moreover, it is shown that these solutions can be converted into Jacobi elliptic function representations, and their long‐wave limit yields collision of dark solitons. In comparing with known solutions of the canonical evolution equation, three new aspects will be developed in this paper. First, the periods in the spatial and temporal directions, measured in terms of the theta function parameters τ and τ₁, are independent of each other, quite unlike most similar solutions found earlier in the literature. Second, the doubly periodic wave solutions possess two families of the local maxima, whose locations and types are then examined in detail. Third, we obtain new doubly periodic standing wave solutions for the Davey–Stewartson equation through its similarity transformation to the coupled Higgs field equation.     

Doubly periodic solutions of the coupled scalar field equations

Some doubly periodic (Jacobi elliptic function) solutions of the coupled Schrödinger–Boussinesq (KdV) equations are presented in closed form. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solution to construct doubly periodic solutions of the coupled equations. When the module m→1, these solutions degenerate to the exact solitary wave solutions of the coupled equations.     

Solitary wave and doubly periodic wave solutions for the Kersten–Krasil’shchik coupled KdV–mKdV system

In this work we devise an algebraic method to uniformly construct solitary wave solutions and doubly periodic wave solutions of physical interest for the Kersten–Krasil’shchik coupled KdV–mKdV system. This system as the classical part of one of superextension of the KdV equation was proposed very recently. The complete integrability, singular analysis and Lax pairs for this system have been found, but its exact solution are still unknown.     

Propagating wave patterns for the ‘resonant’ Davey–Stewartson system

The resonant nonlinear Schrödinger (RNLS) equation exhibits the usual cubic nonlinearity present in the classical nonlinear Schrödinger (NLS) equation together with an additional nonlinear term involving the modulus of the wave envelope. It arises in the context of the propagation of long magneto-acoustic waves in cold, collisionless plasma and in capillarity theory. Here, a natural (2 + 1) (2 spatial and 1 temporal)-dimensional version of the RNLS equation is introduced, termed the ‘resonant’ Davey–Stewartson system. The multi-linear variable separation approach is used to generate a class of exact solutions, which will describe propagating, doubly periodic wave patterns.     

The linear sampling method for the inverse electromagnetic scattering by a partially coated bi‐periodic structure

In this paper, we consider the inverse problem of recovering a doubly periodic Lipschitz structure through the measurement of the scattered field above the structure produced by point sources lying above the structure. The medium above the structure is assumed to be homogeneous and lossless with a positive dielectric coefficient. Below the structure is a perfect conductor partially coated with a dielectric. A periodic version of the linear sampling method is developed to reconstruct the doubly periodic structure using the near field data. In this case, the far field equation defined on the unit ball of ?³ is replaced by the near field equation which is a linear integral equation of the first kind defined on a plane above the periodic surface. Copyright © 2010 John Wiley & Sons, Ltd.     

Exact periodic kink-wave and degenerative soliton solutions for potential Kadomtsev–Petviashvili equation

Exact periodic kink-wave solution, periodic soliton and doubly periodic solutions for the potential Kadomtsev–Petviashvii (PKP) equation are obtained using homoclinic test technique and extended homoclinic test technique, respectively. It is investigated that periodic soliton is degenerated into doubly periodic wave varying with direction of wave propagation.     

New periodic solitary-wave solutions for the Benjiamin Ono equation

In this paper, the new periodic solitary wave and doubly periodic solutions for (1 + 1)-dimensional Benjiamin Ono equation are obtained, using the bilinear method and extended homoclinic test approach. These results demonstrate that the integrable system has richly dynamical behavior even if it is (1 + 1)-dimensional.     

利用F-展开法求解ZK-BBM方程

利用F-展开法求解出了ZK-BBM方程的双周期波解,并在极限形式下得到了ZK-BBM方程的孤波解和单周期波解.从而丰富了该方程解的理论.此方法也可适用求解其它非线性发展方程.     

Backlund transformations and Painleve analysis: Exact solutions for a Grad--Shafranov-type magnetohydrodynamic equilibrium

The problem of plasma equilibrium in a gravitational field isinvestigated analytically. For the two-dimensional problem,the system of ideal magnetohydrodynamic equations is reducedto a single nonlinear elliptic equation of the magnetic potentialas a Grad-Shafranov-type equation. By specifying the arbitraryfunctions in this equation, the sinh-Poisson equation can beobtained. Using the Bäcklund-transformation technique andPainlevé analysis, a set of exact solutions are obtainedwhich adequately describe force-free models for solar flaresand plane-parallel filaments of a diffuse magnetized plasmasuspended horizontally in equilibrium in a uniform gravitationalfield.     

An existence theorem for a generalized self-dual Chern–Simons equation and its application

In this paper, we establish an existence theorem for a generalized self-dual Chern–Simons equation over a doubly periodic domain and use the existence theorem to prove the existence of doubly periodic self-dual vortices in a Maxwell–Chern–Simons model with non-minimal coupling. We find a necessary and sufficient condition for the existence of solutions of the generalized Chern–Simons equation. We prove the existence result by using two methods, a super- and sub-solution method and a constrained minimization method. Our main contribution is that we find a general inequality-type constraint by using the second method and it maybe applied to some related problems with the similar structures.     

Oscillating solitons of the driven,damped nonlinear SchrÖdinger equation

We obtain time-periodic solitons of the parametrically driven, damped nonlinear Schrödinger equation as solutions of the boundary value problem on a two-dimensional domain. We classify the stability and bifurcations of singly and doubly periodic solutions.     

Difference Equation for Functions Analytic Outside Two Squares

We consider a totality of two squares built on primitive periods 1 and i and “sufficiently close to each other“. In a vicinity of this set we investigate four-element difference equation with constant coefficients, whose linear shifts are generating transforms of the corresponding doubly periodic group and the inverse transforms. We seek a solution in a class of functions, which are analytic outside this set and vanish at infinity. The equation is applicable to the moments problem for entire functions of exponential type.     

Travelling wave solutions of the non-linear Schrödinger’s equation in non-Kerr law media

This paper studies the nonlinear Schrödinger’s equation in a non-Kerr law media. The travelling wave ansatz is used to carry out the analysis. The doubly periodic wave solutions are obtained that are supported by numerical simulations.     

Meromorphic solutions of a Riccati differential equation with a doubly periodic coefficient

We treat a Riccati differential equation w^′+w²+p(z)=0, where p(z) is a nonconstant doubly periodic meromorphic function. Under certain assumptions, every solution is meromorphic in the whole complex plane. We show that the growth order of it is equal to 2, and examine the frequency of α-points and poles. Furthermore, the number of doubly periodic solutions is discussed.     

Extend three-wave method for the (1+2)-dimensional Ito equation

In this work, Extend three-wave method (ETM) is used to construct the novel multi-wave solutions of the (1+2)-dimensional Ito equation. As a result, three-soliton solution, doubly periodic solitary wave solutions, periodic two solitary wave solutions are obtained. It is shown that the Extend three-wave method may provide us with a straightforward and effective mathematical tool for seeking multi-wave solutions of higher dimensional nonlinear evolution equations.     

(3+1)-维广义随机KP方程的精确解

利用统一方式构造非线性偏微分方程行波解的广义Jacobi椭圆函数展开法和Hermite变换来研究(3+1)-维广义随机KP方程,给出了它的随机对偶周期和多孤子解.