Jacobian elliptic function expansion solutions for the Wick-type stochastic coupled KdV equations

The Jacobian elliptic function expansion approach is extended and applied to construct exact solutions of the nonlinear Wick-type stochastic coupled KdV equations and some new exact solutions are obtained via the approach and Hermite transformation.     

Symmetry reductions and soliton-like solutions for stochastic MKdV equation

In this paper, the Wick-type stochastic MKdV equation is researched by using symmetry reduction. And some stochastic soliton-like solutions are given via Hermite transformation.     

White noise functional solutions of Wick-type stochastic generalized Hirota–Satsuma coupled KdV equations

Generalized Hirota–Satsuma coupled KdV equations with variable coefficients and Wick-type stochastic generalized Hirota–Satsuma coupled KdV equations are investigated. White noise functional solutions are shown by Hermite transform, homogeneous balance principle and F-expansion method.     

Steady soliton-like solutions of Euler's equations with intrinsic force fields

Steady solutions of a system of Euler's equations when there are intrinsic force interactions of electrical and gravitational type are analysed. Particular attention is devoted to the class of soliton “particle-like” solutions with concentrated charge and mass. Periodic steady solutions for the case when the intrinsic gravitational field has non-zero density at infinity is also considered.     

The smooth solutions of a class of coupled KdV equations.

This paper is devoted to the study of the periodic initial boundary value problem and Cauchy problem for the coupled KdV equations. By the Galerkin method and sequential approximation, we get a seies of priori estimates and establish the existence of classical local solution to the periodic problem for the system. Then we obtain the existence and uniqueness of global smooth solution when the coefficients of the system satisfy certain conditions by energy menthod, conserved quantities and nonconservative quatity I(u,v).     

Some new soliton-like solutions and periodic wave solutions with loop or without loop to a generalized KdV equation

In this paper, by using the integral bifurcation method, we study a generalized KdV equation which was first derived by Fokas from physical considerations via a methodology of Fuchssteiner. All kinds of soliton-like or kink-like wave solutions and periodic wave solutions with loop or without loop are obtained. Smooth compacton-like periodic wave solution and non-smooth periodic cusp wave solution are also obtained. Their dynamic properties are investigated and their profiles are given by Mathematical software.     

Symmetry reductions and soliton-like solutions for the variable coefficient MKdV equations

Two types of symmetry reductions are derived for the variable coefficient MKdV equation, which contain well-known Painleve II type equation and Jacobian elliptic equation. In addition, soliton-like solutions of the variable coefficient MKdV equation are also obtained. Finally, a transformation between the variable coefficient MKdV equation and the MKdV equation are also found.     

Markov solutions of stochastic differential equations

Probability Theory and Related Fields -     

Multiple complex soliton solutions for integrable negative-order KdV and integrable negative-order modified KdV equations

In this work we show that the integrable negative-order Korteweg–de Vries (nKdV) and the integrable negative-order modified Korteweg–de Vries (nMKdV) equation admit multiple complex soliton solutions. To achieve this goal, we introduce two complex forms of the simplified Hirota’s method, the first works effectively for the nKdV equation, and the other form is nicely applicable for the nMKdV equation. We believe that the newly proposed complex forms and the obtained findings will shed light on complex solitons of other integrable equations.     

Exact solutions for a Wick-type stochastic 2D KdV equation

A Wick-type stochastic 2D KdV equation with variable coefficients is investigated. The exact solutions are showed by using the homogeneous balance principle and the Herimite transform in the white noise space.     

Homoclinic breather-wave solutions and doubly periodic wave solutions for coupled KdV equations

In this paper, we first find out a proper variable transformation by the ideas of Painleve expansion. Then we apply the extended homoclinic test approach to obtain two-cycle breathing places wave solutions of Eq. (1) which describe interactions between two physical waves, and these special solutions can be applied to explain the structure of certain physical phenomena. Thus this method can be applied to the study of other similar nonlinear coupled system.     

Martingale solutions for stochastic Euler equations

We prove existence of martingale solutions for the Euler equations in the 2-Dimensional space. The result is obtained by a compactness method     

Strong solutions to stochastic Volterra equations

In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties for stochastic convolutions are studied. Our main results provide sufficient conditions for strong solutions to stochastic Volterra equations.     

Convergence of solutions of backward stochastic equations

We establish conditions for the weak convergence of solutions of backward stochastic equations in the case of the weak convergence of coefficients.     

Lp solutions of backward stochastic differential equations

In this paper, we are interested in solving backward stochastic differential equations (BSDEs for short) under weak assumptions on the data. The first part of the paper is devoted to the development of some new technical aspects of stochastic calculus related to BSDEs. Then we derive a priori estimates and prove existence and uniqueness of solutions in L^p p>1, extending the results of El Karoui et al. (Math. Finance 7(1) (1997) 1) to the case where the monotonicity conditions of Pardoux (Nonlinear Analysis; Differential Equations and Control (Montreal, QC, 1998), Kluwer Academic Publishers, Dordrecht, pp. 503–549) are satisfied. We consider both a fixed and a random time interval. In the last section, we obtain, under an additional assumption, an existence and uniqueness result for BSDEs on a fixed time interval, when the data are only in L¹.     

The uniqueness of solutions of perturbed backward stochastic differential equations

We prove a result on the preservation of the pathwise uniqueness property for the adapted solution to backward stochastic differential equation under perturbations.