Existence and stability of vector solitary waves for nonlinear Schrödinger systems of Hartree-type with Bessel potential

In this paper, we study a class of two-dimensional nonlinear Schrödinger systems of Hartree-type with Bessel potential kernel, which models the propagation and interaction of two-color light beams in nematic liquid crystals. The global well-posedness is proved by using fixed point argument, Gagliardo-Nirenberg inequality and conservation laws. In addition, we also obtain the existence and orbital stability of ground state vector solitary waves applying variational methods and Concentration-compactness Lemma.     

Locally Minimizing Smooth Harmonic Maps from Asymptotically at Manifolds into Spheres

By minimizing the so-called relative energy, we show that there exists a family of locally minimizing smooth harmonic maps from asymptotically flat manifolds into the standard sphere.     

具有非局部反应的时滞扩散Nicholson方程的行波解

对具有扩散项的时滞Nicholson方程的行波解进行了研究．特别是考虑到生物个体在空间位置上的迁移,研究了具有非局部反应的时滞扩散模型．对于弱生成时滞核,运用几何奇异摄动理论,在时滞充分小的情况下,证明了行波解的存在性．     

On travelling waves in a suspension bridge model as the wave speed goes to zero

Travelling waves exist in nonlinearly supported beams. It is shown that as the wave speed goes to zero for a special type of nonlinearity, the amplitude of the waves must go to infinity.     

A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam

For a nonlinear beam equation with exponential nonlinearity, we prove existence of at least 36 travelling wave solutions for the specific wave speed c=1.3. This complements the result in [Smets, van den Berg, Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations, J. Differential Equations 184 (2002) 78-96.] stating that for almost all there exists at least one solution. Our proof makes heavy use of computer assistance: starting from numerical approximations, we use a fixed point argument to prove existence of solutions “close to” the computed approximations.     

Travelling Waves in a Nonlocal Reaction-Diffusion Equation as a Model for a Population Structured by a Space Variable and a Phenotypic Trait

We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed c* > 0, and prove the existence of waves when c ≥ c* and the nonexistence when 0 ≤ c < c*.     

Harmonic Maps and Critical Points of Penalized Energy

We discuss a sequence solutions u_ε for the E-L equations of the penalized energy defined by Chen-Struwe. We show that the blow-up set of u_ε is a H^{m-2} - rectifiable set and its weak limit satisfies a blow-up formula. Consequently, the weak limit will be a stationary harmonic map if and only if the blow-up set is stationary.     

Asymptotic expansions for steady periodic water waves in flows with constant vorticity

We provide high-order approximations to periodic travelling wave profiles, through a novel expansion which incorporates the variation of the total mechanical energy of the water wave. We show that these approximations are extended to any finite order. Moreover, we provide the velocity field and the pressure beneath the waves, in flows with constant vorticity over a flat bed.     

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity: One and Two Dimensional Cases

For N = 1,2, we consider singularly perturbed elliptic equations ?²Δ u ? V(x) u + f(u)= 0, u(x)> 0 on R ^N , lim_|x|→∞ u(x)= 0. For small ? > 0, we show the existence of a localized bound state solution concentrating at an isolated component of positive local minimum of V under conditions on f we believe to be almost optimal; when N ≥ 3, it was shown in Byeon and Jeanjean (2007 Byeon , J. , Oshita , Y. ( 2004 ). Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations . Comm. PDE 29 : 1877 – 1904 . [Google Scholar]).     

New exact travelling wave solutions of Kawahara type equations

The Auxiliary equation method is used to find analytic solutions for the Kawahara and modified Kawahara equations. It is well known that different types of exact solutions of the given auxiliary equation produce new types of exact travelling wave solutions to nonlinear equations. In this paper, new exact solutions of the auxiliary equation are presented. Using these solutions, many new exact travelling wave solutions for the Kawahara type equations are obtained.     

On the Regularity Theories for Harmonic Maps from Finsler Manifolds

The author studies the regularity of energy minimizing maps from Finsler manifolds to Riemannian manifolds.It is also shown that the energy minimizing maps are smooth,when the target manifolds have no ...     

Chern–Simons limit of the standing wave solutions for the Schrödinger equations coupled with a neutral scalar field

We show the existence of standing wave solutions to the Schrödinger equation coupled with a neutral scalar field. We also verify the Chern–Simons limit for these solutions. More precisely we prove that solutions to Eqs. (1.3)–(1.4) converge to the unique positive radially symmetric solution of the nonlinear Schrödinger equation (1.6) as the coupling constant q goes to infinity.