Darboux transformation and Rogue waves of the Kundu–nonlinear Schrödinger equation

In this paper, the Darboux transformation of the Kundu–nonlinear Schrödinger equation is derived and generalized to the matrix of n‐fold Darboux transformation. From known solution Q, the determinant representation of n‐th new solutions of Q^[n] are obtained by the n‐fold Darboux transformation. Then soliton solutions and positon solutions are generated from trivial seed solutions, breather solutions and rogue wave solutions that are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu–nonlinear Schrödinger equation are given. We show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves. Meanwhile, the third‐order rogue waves are given explicitly. Copyright © 2014 John Wiley & Sons, Ltd.     

Rational and semi‐rational solutions of a nonlocal (2 + 1)‐dimensional nonlinear Schrödinger equation

We consider the fully parity‐time (PT) symmetric nonlocal (2 + 1)‐dimensional nonlinear Schrödinger (NLS) equation with respect to x and y. By using Hirota's bilinear method, we derive the N‐soliton solutions of the nonlocal NLS equation. By using the resulting N‐soliton solutions and employing long wave limit method, we derive its nonsingular rational solutions and semi‐rational solutions. The rational solutions act as the line rogue waves. The semi‐rational solutions mean different types of combinations in rogue waves, breathers, and periodic line waves. Furthermore, in order to easily understand the dynamic behaviors of the nonlocal NLS equation, we display some graphics to analyze the characteristics of these solutions.     

All meromorphic solutions of an ordinary differential equation and its applications

Dedicated to Professor Yuzan He on the Occasion of his 80th Birthday In this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first and then find out all meromorphic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations. Our result shows that all rational and simply periodic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations are solitary wave solutions, the method is more simple than other methods, and there exist some rational solutions w_r,2(z) and simply periodic solutions w_s,2(z) that are not only new but also not degenerated successively by the elliptic function solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability Results for Efficient Solutions of Vector Optimization Problems

Using the additive weight method of vector optimization problems and the method of essential solutions, we study some continuity properties of the mapping which associates the set of efficient solutions S(f) to the objective function f. To understand such properties, the key point is to consider the stability of additive weight solutions and the relationship between efficient solutions and additive weight solutions.     

Bifurcations of travelling wave solutions for the mK(n, n) equation

Using the method of planar dynamical systems to the mK(n, n) equation, the existence of uncountably infinite many smooth and non-smooth periodic wave solutions, solitary wave solutions and kink and anti-kink wave solutions is proved. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All possible exact explicit parametric representations of smooth and non-smooth travelling wave solutions are obtain.     

Population monotonic solutions on convex games

The Dutta-Ray solution and the Shapley value are two well-known examples of population-monotonic solutions on the domain of convex games. We provide a new formula for the Dutta-Ray solution from which population-monotonicity immediately follows. Then we define a new family of population-monotonic solutions, which we refer to as “sequential Dutta-Ray solutions.” We also show that it is possible to construct several symmetric and population-monotonic solutions by using the solutions in this family. Received September 1998/Revised version: December 1999     

Darboux Transformations and Global Explicit Solutions for Nonlocal Davey–Stewartson I Equation

For the nonlocal Davey–Stewartson I equation, the Darboux transformation is considered and explicit expressions of the solutions are obtained. Like other nonlocal equations, many solutions of this equation may have singularities. However, by suitable choice of parameters in the solutions of the Lax pair, it is proved that the solutions obtained from seed solutions which are zero and an exponential function of t , respectively, by a Darboux transformation of degree n are global solutions of the nonlocal Davey–Stewartson I equation. The derived solutions are soliton solutions when the seed solution is zero, in the sense that they are bounded and have n peaks, and “dark cross soliton” solutions when the seed solution is an exponential function of t , in the sense that they are bounded and their norms change fast along some crossing straight lines.     

Periodic and boundary value problems for second order differential equations

In this paper we study second order scalar differential equations with Sturm-Liouville and periodic boundary conditions. The vector fieldf(t,x,y) is Caratheodory and in some instances the continuity condition onx ory is replaced by a monotonicity type hypothesis. Using the method of upper and lower solutions as well as truncation and penalization techniques, we show the existence of solutions and extremal solutions in the order interval determined by the upper and lower solutions. Also we establish some properties of the solutions and of the set they form.     

All traveling wave exact solutions of three kinds of nonlinear evolution equations

In this article, we employ the complex method to obtain all meromorphic exact solutions of complex Klein–Gordon (KG) equation, modified Korteweg‐de Vries (mKdV) equation, and the generalized Boussinesq (gB) equation at first, then find all exact solutions of the Equations KG, mKdV, and gB. The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic solutions are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions w_2r,2(z) and simply periodic solutions w_1s,2(z),w_2s,1(z) in these equations such that they are not only new but also not degenerated successively by the elliptic function solutions. We have also given some computer simulations to illustrate our main results. Copyright © 2014 John Wiley & Sons, Ltd.     

Lq‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate L^q‐estimates. The uniqueness of very weak solutions is shown by the method of cut‐off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D^1,r‐result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D^1,r is the homogeneous Sobolev space.     

Spectral Analysis for Stationary Solutions of the Cahn–Hilliard Equation in ℝ d

We consider the spectrum associated with three types of bounded stationary solutions for the Cahn–Hilliard equation on ? ^d , d ≥ 2: radial solutions, saddle solutions (only for d = 2), and planar periodic solutions. In particular, we establish spectral instability for each type of solution. The important case of multiply periodic solutions does not fit into the framework of our approach, and we do not consider it here.     

Prolongation of solutions of measure differential equations and dynamic equations on time scales

In this paper, we prove the results on existence and uniqueness of the maximal solutions for measure differential equations, considering more general conditions on functions f and g by using the correspondence between the solutions of these equations and the solutions of generalized ODEs. Moreover, we prove these results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations.     

DISCONTINUOUS SOLUTIONS IN L^∞ FOR HAMILTON-JACOBI EQUATIONS

An approach is introduced to construct global discontinuous solutions in L∞ for Hamilton-Jacobi equations. This approach allows the initial data only in L∞ and applies to the equations with nonconvex Hamiltonians. The profit functions are introduced to formulate the notion of discontinuous solutions in L∞. The existence of global discontinuous solutions in L∞ is established. These solutions in L∞ coincide with the viscosity solutions and the minimax solutions, provided that the initial data are continuous. A prototypical equation is analyzed toexamine the L∞ stability of our L∞ solutions. The analysis also shows that global discontinuous solutions are determined by the topology in which the initial data are approximated.     

Well posedness of sudden directional diffusion equations

Our goal is to establish existence with suitable initial data of solutions to general parabolic equation in one dimension, u_t = L(u_x)_x, where L is merely a monotone function. We also expose the basic properties of solutions, concentrating on maximal possible regularity. Analysis of solutions with convex initial data explains why we may call them almost classical. Some qualitative aspects of solutions, such as facets (flat regions of solutions), are studied too. Copyright © 2013 John Wiley & Sons, Ltd.     

Liouville theorems for the Navier–Stokes equations and applications

We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in R ⁿ × (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].     

New exact solutions for the classical Drinfel’d–Sokolov–Wilson equation

In this paper, we investigate the classical Drinfel’d–Sokolov–Wilson equation (DSWE)

where p, q, r, s are some nonzero parameters. Some explicit expressions of solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, blow-up solutions, periodic solutions, periodic blow-up solutions and kink-shaped solutions. Some previous results are extended.     

Hölder continuity of solutions to a degenerate elliptic equation in the presence of the Lavrentiev phenomenon

We study a second-order elliptic equation for which the Dirichlet problem can be posed in a nonunique way due to the so-called Lavrentiev phenomenon. In the corresponding weighted Sobolev space smooth functions are not dense, which leads to the existence of W – solutions and H – solutions. For H - solutions, we establish the Hölder continuity. We also discuss this question for W – solutions, for which the situation is more complicated.     

Infinitely many solutions of a second-order <Emphasis Type="Italic">p</Emphasis>-Laplacian problem with impulsive condition

Using the critical point theory and the method of lower and upper solutions, we present a new approach to obtain the existence of solutions to a p-Laplacian impulsive problem. As applications, we get unbounded sequences of solutions and sequences of arbitrarily small positive solutions of the p-Laplacian impulsive problem.     

The entire solutions of a polynomial difference equation

Recently S. Shimomura has shown that the polynomial difference equationw(z + 1) =P(w(z)), whereP is a given polynomial of degree at least two, always has entire non-constant solutions. The present investigation shows how to construct all entire solutions of the equation and discusses some properties of the solutions.