Exact travelling wave solutions for the dissipative (2 + 1)-dimensional AKNS equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated. In addition, the possible bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on above studies, a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation.     

Self-adjointness of a generalized Camassa-Holm equation

It is well known that the Camassa-Holm equation possesses numerous remarkable properties characteristic for KdV type equations. In this paper we show that it shares one more property with the KdV equation. Namely, it is shown in [1] and [2] that the KdV and the modified KdV equations are self-adjoint. Starting from the generalization [3] of the Camassa-Holm equation [4], we prove that the Camassa-Holm equation is self-adjoint. This property is important, e.g. for constructing conservation laws associated with symmetries of the equation in question. Accordingly, we construct conservation laws for the generalized Camassa-Holm equation using its symmetries.     

Monotonic positive solution of a nonlinear quadratic functional integral equation

The existence of positive monotonic solutions, in the class of continuous functions, for some nonlinear quadratic integral equation have been studied in [4], [5], [6], [7] and [8]. Here we are concerning with a nonlinear quadratic integral equation of Volterra type and we shall prove the existence of at least one L₁-positive monotonic solution for that equation under Carathèodory condition.     

Dynamical bifurcation for the Kuramoto-Sivashinsky equation

In this paper, by using the center manifold reduction method, together with the eigenvalue analysis, we made bifurcation analysis for the Kuramoto-Sivashinsky equation, and proved that the Kuramoto-Sivashinsky equation with constraint condition bifurcates an attractor A_λ as λ crossed the first critical value λ₀=1 under the two cases. Our analysis was based on a new and mature attractor bifurcation theory developed by Ma and Wang (2005) [17] and [18].     

Exact traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation

Bifurcation method of dynamical systems is employed to investigate traveling wave solutions in the (2 + 1)-dimensional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation. Under some parameter conditions, exact solitary wave solutions and kink wave solutions are obtained.     

Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3d ball

We establish new global well-posedness results along Gibbs measure evolution for the nonlinear wave equation posed on the unit ball in R³${\mathbb{R}}^3$ via two distinct approaches. The first approach invokes the method established in the works Bourgain (1994, 1996) ,  and  based on a contraction-mapping principle and applies to a certain range of nonlinearities. The second approach allows to cover the full range of nonlinearities admissible to treatment by Gibbs measure, working instead with a delicate analysis of convergence properties of solutions. The method of the second approach is quite general, and we shall give applications to the nonlinear Schrödinger equation on the unit ball in subsequent works Bourgain and Bulut (2013)  and .     

Generalized solitary and periodic wave solutions to a (2 + 1)-dimensional Zakharov-Kuznetsov equation

In this paper, the Exp-function method is employed to the Zakharov-Kuznetsov equation as a (2 + 1)-dimensional model for nonlinear Rossby waves. The observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient. The obtained results may be useful for better understanding the properties of two-dimensional coherent structures such as atmospheric blocking events.     

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.     

A variety of exact travelling wave solutions for the (2 + 1)-dimensional Boiti-Leon-Pempinelli equation

In this paper the (2 + 1)-dimensional Boiti-Leon-Pempinelli (BLP) equation will be studied. The tanh-coth method will be used to obtain exact travelling wave solutions for this equation. The Exp-function method will also be applied to the BLP equation to derive a new variety of travelling wave solutions with distinct physical structures.     

Explicit solutions of (2 + 1)-dimensional nonlinear KP-BBM equation by using Exp-function method

This paper is devoted to studying the (2 + 1)-dimensional KP-BBM wave equation. Exp-function method is used to conduct the analysis. The generalized solitary solutions, periodic solutions and other exact solutions for the (2 + 1)-dimensional KP-BBM wave equation are obtained via this method with the aid of symbolic computational system. It is also shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.     

Differential operator related to the generalized superradiance integral equation

In this paper we consider a class of problems which are generalized versions of the three-dimensional superradiance integral equation. A commuting differential operator will be found for this generalized problem. For the three-dimensional superradiance problem an alternative set of complete eigenfunctions will also be provided. The kernel for the superradiance problem when restricted to one-dimension is the same as appeared in the works of Slepian, Landau and Pollak (cf. Slepian and Pollak (1961) [1], Landau and Pollak (1961, 1962) [2] and [3], Slepian (1964, 1978) [4] and [5]). The uniqueness of the differential operator commuting with that kernel is indicated.     

New exact periodic solitary wave solutions for Kadomtsev-Petviashvili equation

Exact periodic solitary wave solutions for Kadomtsev-Petviashvili equation are obtained by using the Hirota bilinear method. The result shows that there exists periodic solitary waves in the different directions for (2 + 1)-dimensional Kadomtsev-Petviashvili equation.     

Peakon, loop and solitary travelling wave solutions for the general modified CH-DP equation

Travelling wave solutions for the general modified CH-DP equation u_t − u_xxt + αu²u_x − βu_xu_xx = uu_xxx are developed. By using the dynamical system method, a peakon and a dark soliton are found to coexist for the same wave speed. Exact explicit blow-up solutions are given. By using numerical simulation, a loop solution for a special case is discussed.     

Determining surface heat flux in the steady state for the Cauchy problem for the Laplace equation

In this paper, we consider the Cauchy problem for the Laplace equation, in a strip where the Cauchy data is given at x = 0 and the flux is sought in the interval 0<x?1. This problem is typical ill-posed: the solution (if it exists) does not depend continuously on the data. We study a modification of the equation, where a fourth-order mixed derivative term is added. Some error stability estimates for the flux are given, which show that the solution of the modified equation is approximate to the solution of the Cauchy problem for the Laplace equation. Furthermore, numerical examples show that the modified method works effectively.     

Two-periodic waves and asymptotic property for generalized 2D Toda lattice equation

In this paper, two-periodic wave solutions are constructed for the (2 + 1)-dimensional generalized Toda lattice equation by using Hirota bilinear method and Riemann theta function. At the same time, we analyze in details asymptotic properties of the two-periodic wave solutions and give their asymptotic relations between the periodic wave solutions and the soliton solutions.     

Interactions of solitary waves under the conditions of the (3 + 1)-dimensional Kadomtsev-Petviashvilli equation

Using an extended mapping method with a linear variable separation process, a new family of the exact solutions of the (3 + 1)-dimensional Kadomtsev-Petviashvilli (KP) equation was derived. By applying the solitary wave solutions, this paper studied some newly localized excitations and the interactions of various solitary waves under the conditions of the (3 + 1)-dimensional KP equation.     

On the dynamics of a strongly singular parabolic equation

In this paper we study the semiflow defined by a semilinear parabolic equation, in which both the diffusion and the reaction term present strong order of singularity at the origin. We justify the existence of a global branch of nontrivial equilibrium solutions for subcritical nonlinearities. The approach, based on the critical Caffarelli–Kohn–Nirenberg inequality, follows the arguments of  and .     

Explicit periodic solitary wave solutions for the (2 + 1)-dimensional Boussinesq equation

In this paper, (2 + 1)-dimensional Boussinesq equation is investigated. By using homoclinic test method with the aid of Maple, new explicit periodic solitary wave solutions are obtained. Moreover, mechanical feature of wave is exhibited.     

Explicit solutions of the (2 + 1)-dimensional AKNS shallow water wave equation with variable coefficients

Using an improved direct reduction method, we find the equivalence transformations of (2 + 1)-dimensional AKNS shallow water wave equation with variable coefficients, and obtain the corresponding relationship between explicit solutions of AKNS equation and those of the corresponding reduced equation. In addition, we get some new explicit solutions of AKNS equation by applying Lie symmetry method.     

On a stochastic differential equation arising in a price impact model

We provide sufficient conditions for the existence and uniqueness of solutions to a stochastic differential equation which arises in the price impact model developed by Bank and Kramkov (2011)  and . These conditions are stated as smoothness and boundedness requirements on utility functions or Malliavin differentiability of payoffs and endowments.