Related papers: Homoclinic solutions for fourth order traveling wa…

We present a representation of skew-orthogonal polynomials of symplectic type ($\beta=4$) in terms of a matrix Riemann-Hilbert problem, for weights of the form ${\rm e}^{-V(z)}$ where $V$ is a polynomial of even degree and positive leading…

We study the existence of traveling waves for the parabolic system \begin{equation} \partial_t w - \partial_{x}^2 w = -\nabla_{\mathbb{u}} W(w) \mbox{ in } [0,+\infty) \times \mathbb{R} \end{equation} where $W$ is a potential bounded below…

We show stabilisation of solutions to the sixth-order convective Cahn-Hilliard equation. {The problem} has the structure of a gradient flow perturbed by a quadratic destabilising term with coefficient $\delta>0$. Through application of an…

Exact stationary soliton solutions of the fifth order KdV type equation $$ u_t +\alpha u^p u_x +\beta u_{3x}+\gamma u_{5x} = 0$$ are obtained for any p ($>0$) in case $\alpha\beta>0$, $D\beta>0$, $\beta\gamma<0$ (where D is the soliton…

By variational methods, we provide a simple proof of existence of a heteroclinic orbit to the Hamiltonian system $u''=\nabla W(u)$ that connects the two global minima of a double-well potential $W$. Moreover, we consider several…

This paper is devoted to the existence of positive solutions for a problem related to a fourth-order differential equation involving a nonlinear term depending on a second order differential operator, $$(-\Delta)^2 u=\lambda u+…

We consider the self-dual Chern-Simons-Schr\"odinger equation (CSS). CSS is $L^{2}$-critical, admits solitons, and has the pseudoconformal symmetry. In this work, we consider pseudoconformal blow-up solutions under $m$-equivariance,…

In this paper, we are concerned with static Schr\"{o}dinger-Hartree and Schr\"{o}dinger-Maxwell equations with combined nonlinearities. We derive the explicit forms for positive solution $u$ in the critical case and non-existence of…

We study the existence of fixed points to a parameterized Hammertstain operator $\cH_\beta,$ $\beta\in (0,\infty],$ with sigmoid type of nonlinearity. The parameter $\beta<\infty$ indicates the steepness of the slope of a nonlinear smooth…

We obtain sufficient conditions expressed in terms of Wiener type tests involving Hausdorff or Bessel capacities for the existence of large solutions to equations (1) $-\Gd_pu+e^{\lambda u}+\beta=0$ or (2) $-\Gd_pu+\lambda…

Motivated by the viewpoint of integrable systems, we study commuting flows of 2-component quasilinear equations, reducing to investigate the solutions of the wave equation with non-constant speed. In this paper, we apply the reduction…

We examine the fourth order problem $\Delta^2 u = \lambda f(u) $ in $ \Omega$ with $ \Delta u = u =0 $ on $ \partial \Omega$, where $ \lambda > 0$ is a parameter, $ \Omega$ is a bounded domain in $ R^N$ and where $f$ is one of the following…

A new solution to the Einstein-Maxwell field equations is presented describing a cylindrically symmetric homogeneous cosmology. The solution is conformally flat, it possesses seven Killing vectors of which the timelike one is rotating and…

Fourth-order semilinear parabolic equations of the Cahn--Hilliard-type (01) u_t + \D^2 u = \g u \pm \D (|u|^{p-1}u) in \Omega \times \re_+, are considered in a smooth bounded domain $\O \subset \ren$ with Navier-type boundary conditions on…

We consider the conservative complex Swift-Hohenberg equation, which belongs to the family of nonlinear fourth-order dispersive Schr\"odinger equations. In contrast to the well-studied one-dimensional dissipative Swift-Hohenberg equation,…

Given $n \geq 2$ and $1<p<n$, we consider the critical $p$-Laplacian equation $\Delta_p u + u^{p^*-1}=0$, which corresponds to critical points of the Sobolev inequality. Exploiting the moving planes method, it has been recently shown that…

In this paper we prove the existence and uniqueness of positive mild solutions for the semilinear parabolic equations of the form $u_t+\mathcal{L}u=f+h\cdot G(u)$, where $h$ is a positive function and $G$ a positive concave function (for…

Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function. Let $H(x,p,u)$ be a continuous Hamiltonian which is strictly…

Exponential small splitting of separatrices in the singular perturbation theory leads generally to nonvanishing oscillations near a saddle--center point and to nonexistence of a true homoclinic orbit. It was conjectured long ago that the…

We study the existence-uniqueness of solution $(u, \lambda)$ to the ergodic Hamilton-Jacobi equation $$(-\Delta)^s u + H(x, \nabla u) = f-\lambda\quad \text{in}\; \mathbb{R}^d,$$ and $u\geq 0$, where $s\in (\frac{1}{2}, 1)$. We show that…