Related papers: Initial boundary value problem for the focusing NL…
Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based…
This paper is concerned with the initial-boundary value problem on the full Euler-Poisson system for ions over a half line. We establish the existence of stationary solutions under the Bohm criterion similar to the isentropic case and…
In this work, we study the initial boundary value problem for a non-strictly hyperbolic $2\times2$ system of equations in the quarter plane $x>0,t>0$ which is derived from Eulerian droplet model for air particle flow for velocity and volume…
We consider the one-dimensional focusing nonlinear Schr\"odinger equation (NLS) with a delta potential and even initial data. The problem is equivalent to the solution of the initial/boundary problem for NLS on a half-line with Robin…
We consider the Gerdjikov--Ivanov type derivative nonlinear Schr\"odinger equation \berr \ii q_{t}+q_{xx}-\ii q^2\bar{q}_{x}+\frac{1}{2}(|q|^4-q_0^4)q=0 \eerr on the line. The initial value $q(x,0)$ is given and satisfies the symmetric,…
In this paper, we apply $\overline\partial$ steepest descent method to study the Cauchy problem for the derivative nonlinear Schr\"odinger equation with nonzero boundary conditions \begin{align} &iq_{t}+q_{xx}+i\sigma(|q|^2q)_{x}=0,\\ &…
This paper studies an initial boundary value problem for a class of nonlinear Dirac equations with cubic terms and moving boundary. For the initial data with bounded $L^2$ norm and the suitable boundary conditions, the global existence and…
In this paper we consider the initial boundary value problem (IBVP) for the nonlinear biharmonic Schr\"odinger equation posed on a bounded interval $(0,L)$ with non-homogeneous Navier or Dirichlet boundary conditions, respectively. For…
The Cauchy- and periodic boundary value problem for the nonlinear Schroedinger equations in $n$ space dimensions [u_t - i\Delta u = (\nabla \bar{u})^{\beta}, |\beta|=m \ge 2, u(0)=u_0 \in H^{s+1}_x] is shown to be locally well posed for $s…
The initial value problem for some defocusing coupled nonlinear Schrodinger equations is investigated. Global well-posedness and scattering are established.
We study the inverse Robin problem for the Schr\"odinger equation in a half-space. The potential is assumed to be compactly supported. We first solve the direct problem for dimensions two and three. We then show that the Robin-to-Robin map…
We consider solutions of the defocusing nonlinear Schr\"odinger (NLS) equation on the half-line whose Dirichlet and Neumann boundary values become periodic for sufficiently large $t$. We prove a theorem which, modulo certain assumptions,…
We consider the Schr\"odinger equation with nonlinear derivative term on $[0,+\infty)$ under Robin boundary condition at $0$. Using a virial argument, we obtain the existence of blowing up solutions and using variational techniques, we…
The characteristic initial boundary problem is discussed in spherical symmetry for the Einstein-Maxwell-scalar field equations. It is formulated for an affine-null metric and the resulting field equations are cast into a hierarchical system…
This paper deals with the initial-boundary value problem of the biharmonic cubic nonlinear Schr\"odinger equation in a quarter plane with inhomogeneous Dirichlet-Neumann boundary data. We prove local well-posedness in the low regularity…
We prove existence and uniqueness of solutions to the initial-boundary value problem for the Lifshitz--Slyozov equation (a nonlinear transport equation on the half-line), focusing on the case of kinetic rates with unbounded derivative at…
We consider solutions of the defocusing nonlinear Schr\"odinger equation in the quarter plane whose Dirichlet boundary data approach a single exponential $\alpha e^{i\omega t}$ as $t \to \infty$. In order to determine the long time…
Based on the theory of integrable boundary conditions (BCs) developed by Sklyanin, we provide a direct method for computing soliton solutions of the focusing nonlinear Schr\"odinger (NLS) equation on the half-line. The integrable BCs at the…
This paper studies the initial-boundary-value problem (IBVP) of a nonlinear Schr\"odinger equation posed on a strip domain $\mathbb{R}\times[0,1]$ with non-homogeneous Dirichlet boundary conditions. For any $s\ge0$, if the initial data…
We explore the phenomena of absorption/emission of solitons by an integrable boundary for the nonlinear Schr\"odinger equation on the half-line. This is based on the investigation of time-dependent reflection matrices which satisfy the…