Related papers: The derivative nonlinear Schrodinger equation on t…
Recently, an integrable system of coupled (2+1)-dimensional nonlinear Schrodinger (NLS) equations was introduced by Fokas (eq. (18) in Nonlinearity 29}, 319324 (2016)). Following this pattern, two integrable equations [eqs.2 and 3] with…
We analytically solved the higher-order nonlinear Schrodinger (HNLS) equation with non-Kerr nonlinearity under some parametric conditions and investigated explicitly bright and dark solitary wave solutions. Periodic wave solutions are also…
We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and…
We consider the nonlinear stationary Schr\"odinger equation \begin{equation*} -\Delta u -\lambda u= Q(x)|u|^{p-2}u, \qquad \text{in }\mathbb{R}^N \end{equation*} in the case where $N \geq 3$, $p$ is a superlinear, subcritical exponent, $Q$…
An integrable generalization of the NLS equation is presented, in which the dynamical complex variable $u(t,x)$ is replaced by a pair of dynamical complex variables $(u_1(t,x),u_2(t,x))$, and $i$ is replaced by a Pauli matrix $J$.…
A nonlocal-in-time problem for the abstract Schr\"odinger equation is considered. By exploiting the linear nature of nonlocal condition we derive an exact representation of the solution operator under assumptions that the spectrum of…
In this work, the higher-order dispersive nonlinear Schr\"{o}dinger equation with non-zero boundary conditions at infinity is investigated including the simple and double zeros of the scattering coefficients. We introduce a appropriate…
In this paper we investigate the coupled focusing-defocusing complex short pulse equation, which describe the propagation of ultra-short optical pulses in cubic nonlinear media. Through the unified transform method, the initial-boundary…
In this paper, we propose a new integrable fractional Fokas--Lenells equation by using the completeness of the squared eigenfunctions, dispersion relation, and inverse scattering transform. To solve this equation, we employ the…
We consider the defocusing quintic nonlinear Schr\"odinger equation in four space dimensions. We prove that any solution that remains bounded in the critical Sobolev space must be global and scatter. We employ a space-localized interaction…
In this paper we analyze the global existence of classical solutions to the initial boundary-value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a…
The paper deals with standing wave solutions of the dimensionless nonlinear Schr\"odinger equation \label{eq:abs1} i\Phi_t(x,t) = -\Delta_x\Phi +V_\la(x)\Phi + f(x,\Phi), \quad x\in\R^N,\ t\in\R,\tag{$NLS_\la$} where the potential…
In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on the finite interval with $3 \times 3$ Lax pairs. The solution can be expressed in terms of…
We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of Schr\"odinger equations in Fock space: the strong resolvent limit is unitary equivalent to…
In this paper, we study a Schr\"odinger-type equation featuring a derivative in the nonlinear term and incorporating diffusion effects. This type of equation arises in various physical applications, such as modeling low-order magnetization…
This work investigates the long time asymptotic behavior of some inhomogeneous non-linear Schr\"odinger type equations. We give sharp a threshold of scattering versus non-scattering of mass solutions, depending on the source term. This work…
Exact expressions are obtained for a diversity of propagating patterns for a derivative nonlinear Schr\"odinger equation with a quintic nonlinearity. These patterns include bright pulses, fronts and dark solitons. The evolution of the wave…
The Dbar dressing method is extended to study the focusing/defocusing nonlinear Schr\"odinger (NLS) equation with nonzero boundary condition. A special type of complex function is considered. The function is meromorphic outside an annulus…
In this paper, we explore the initial-boundary value (IBV) problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on the finite interval by extending the Fokas unified transform approach. The solution of this system…
We study the focusing NLS equation in $\mathbb{R}^N$ in the mass-supercritical and energy-subcritical (or intercritical) regime, with $H^1$ data at the mass-energy threshold $ \mathcal{ME}(u_0)=\mathcal{ME}(Q)$, where $Q$ is the ground…