Class Of Nonlinear Schrodinger Equations Research Articles

TWO-DIMENSIONAL NONLOCAL NONLINEAR SCHRODINGER EQUATION BASED ON THE ABLOWITZ-MUSSLIMANI SYMMETRY CONDITION

The nonlinear Schrodinger equation is a nonlinear partial differential equation and integrable equation that play an essential role in many branches of physics as nonrelativistic quantum mechanics, acoustics, and optics. In this work, motivated by the ideas of Ablowitz and Musslimani, we successfully obtain a two-dimensional nonlocal nonlinear Schrodinger equation where the nonlocality consists of reverse time fields as factors in the nonlinear terms. The nonlocal nonlinear Schrodinger equation admits a great number of good properties that the classical nonlinear Schrodinger equation possesses, e.g. PT-symmetric, admitting Lax-pair, and infinitely many conservation laws. We apply the Darboux transformation method to the two-dimensional nonlinear Schrodinger equation. The idea of this method is having a Lax representation, one can obtain various kinds of solutions of the Nth order with a spectral parameter. The exact solutions and graphical representation of obtained solutions are derived.

Existence of a ground state and blowup problem for a class of nonlinear Schrödinger equations involving mass and energy critical exponents

In this paper, we study the existence of the ground state and blowup problem for a class of nonlinear Schrodinger equations involving the mass and energy critical exponents. To show the existence of a ground state, we solve a minimization problem related to the virial identity, so that we need to compare the minimization value to the best constant of the Gagliardo–Nirenberg inequality because our nonlinearities contain the mass critical nonlinearity. Once we obtain the ground state, we can introduce a subset $${\mathcal {A}}_{\omega , -}$$ of $$H^{1}({\mathbb {R}}^d)$$ for each $$\omega > 0$$ as in Berestycki and Cazenave (C R Acad Sci Paris Ser I Math 293:489–492, 1981). Then, it turn out that any radial solution starting from $${\mathcal {A}}_{\omega , -}$$ blows up in a finite time.

Existence of Solutions to Nonlinear Schrödinger Equations Involving N-Laplacian and Potentials Vanishing at Infinity

We study the existence of solutions for the following class of nonlinear Schrodinger equations −Δnu + V(x)u = K(x)f(u)in ℝN where V and K are bounded and decaying potentials and the nonlinearity f(s) has exponential critical growth. The approaches used here are based on a version of the Trudinger-Moser inequality and a minimax theorem.

Extended generalized Darboux transformation to hybrid rogue wave and breather solutions for a nonlinear Schrödinger equation

Abstract An extended generalized Darboux transformation method is proposed to construct the hybrid rogue wave and breather solutions for a classical nonlinear Schrodinger equation. Three types of hybrid wave solutions are obtained: (i) the hybrid first-order rogue wave and breather; (ii) the hybrid second-order rogue wave and first-order breather; (iii) the hybrid first-order rogue wave and second-order breather. These solutions are novel and can be used to investigate the dynamical characteristic of the hybrid rogue waves and breathers. The control and interaction based on the parameters of the hybrid wave solution are graphically demonstrated. An exact link is established between the hybrid solutions and the rogue wave solutions via setting the parameter at special value.

Exact solution of perturbed nonlinear Schrödinger equation using ($$G^\prime /$$G, 1/G)-expansion method

By constructing auxiliary equations and combining the expansion method of ( $$G^{\prime }/G,1/G$$ ), we study a class of nonlinear Schrodinger equation with perturbation terms which describes the propagation of the waves in optical metamaterials. More types of exact solutions, particularly solitary wave solutions, are obtained for the first time.

ON KINK AND ANTI-KINK WAVE SOLUTIONS OF SCHRÖDINGER EQUATION WITH DISTRIBUTED DELAY

This paper deals with existence problem of traveling wave solutions of a class of nonlinear Schrodinger equation having distributed delay with a strong generic kernal. By using the geometric singular perturbation theory and the Melnikov function method, we establish results of the existence of kink and anti-kink wave solutions of the nonlinear Schrodinger equation with time delay when the average delay is sufficiently small.

Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth

In this paper, we study a class of nonlinear Schrodinger equations involving the fractional Laplacian and the nonlinearity term with critical Sobolev exponent. We assume that the potential of the equations includes a parameter $λ$. Moreover, the potential behaves like a potential well when the parameter λ is large. Using variational methods, combining Nehari methods, we prove that the equation has a least energy solution which, as the parameter λ large, localizes near the bottom of the potential well. Moreover, if the zero set int \begin{document} $V^{-1}(0)$ \end{document} of \begin{document} $V(x)$ \end{document} includes more than one isolated component, then \begin{document} $u_\lambda (x)$ \end{document} will be trapped around all the isolated components. However, in Laplacian case when \begin{document} $s=1$ \end{document} , for \begin{document} $\lambda$ \end{document} large, the corresponding least energy solution will be trapped around only one isolated component and will become arbitrary small in other components of int \begin{document} $V^{-1}(0)$ \end{document} . This is the essential difference with the Laplacian problems since the operator \begin{document} $(-Δ)^{s}$ \end{document} is nonlocal.

On multi-bump solutions of nonlinear Schrödinger equation with electromagnetic fields and critical nonlinearity in $$\mathbb {R}^N$$ R N

In this paper, we study a class of nonlinear Schrodinger equations with electromagnetic fields and critical nonlinearity in \(\mathbb {R}^N: -\Delta _Au + (\lambda V(x) + Z(x))u = \beta f(|u|^2)u + |u|^{2^*-2}u\), where f is a continuous function, \(V, Z: \mathbb {R}^N \rightarrow \mathbb {R}\) are continuous functions verifying suitable hypotheses. We show that if the zero set of V(x) has several isolated connected components \(\Omega _1,\ldots ,\Omega _k\) such that the interior of \(\Omega _i\) is not empty and \(\partial \Omega _i\) is smooth, then for \(\lambda > 0\) large there exists, for any non-empty subset \(\Gamma \subset \{1,\ldots ,k\}\), a bump solution trapped in a neighborhood of \(\cup _{j\in \Gamma }\Omega _j\).

Radially symmetric sign-definite solutions for a class of nonlocal Schrödinger equations

We study a class of nonlinear Schrödinger equations driven by the infinitesimal generators of some general rotationally invariant Lévy processes and with general nonlinearity. Regularity, sign-preserving property, and radial symmetry are shown for the solutions to the Schrödinger equations.

Least energy solutions for nonlinear Schrödinger equations involving the half Laplacian and critical growth

In this paper, we study a class of nonlinear Schrodinger equations involving the half Laplacian and critical growth. We assume that the potential of the nonlinear Schrodinger equation includes a parameter \({\lambda}\). Moreover, the potential behaves like a potential well when the parameter \({\lambda}\) is large. Using variational methods, combining Nehari methods, we prove that the equation has a least energy solution which, for \({\lambda}\) large, localizes near the bottom of the potential well. Moreover, if the zero set int \({V^{-1}(0)}\) of \({V(x)}\) includes more than one isolated component, then \({u_{\lambda}(x)}\) will be trapped around all the isolated components. However, in Laplacian case when \({s = 1}\), for \({\lambda}\) large, the corresponding least energy solution will be trapped around only one isolated component and will become arbitrary small in other components of int \({V^{-1}(0)}\). This is the essential difference with the Laplacian problems since the operator \({(- \Delta)^{1/2}}\) is nonlocal.

Dynamics of matter-wave solitons in Bose-Einstein condensates with time-dependent scattering length and complex potentials

We investigate the dynamics of matter-wave solitons in the one-dimensional (1-D) Gross-Pitaevskii (GP) equation describing Bose-Einstein condensates (BECs) with time-dependent scattering length in varying trapping potentials with feeding/loss term. By performing a modified lens-type transformation, we reduce the GP equation into a classical nonlinear Schrodinger (NLS) equation with distributed coefficients and find its integrable condition. Under the integrable condition, we apply the generalized Jacobian elliptic function method (GJEFM) and present exact analytical solutions which describe the propagation of a bright and dark solitons in BECs. Their stability is examined using analytic method. The obtained exact solutions show that the amplitude of bright and dark solitons depends on the scattering length, while their motion and the total number of BEC atoms depend on the external trapping potential. Our results also shown that the loss of atoms can dominate the aggregation of atoms by the attractive interaction, and thus the peak density can decrease in time despite that the strength of the attractive interaction is increased.

Analytic Self-Consistent Condensates in Quasi-1D Superfluid Fermi Gases in the Andreev Approximation

We present an analytic method to approach Eilenberger equation and the associated Bogoliubov–de Gennes equation for quasi-1D fermionic gases. The problem of finding self-consistent inhomogeneous condensates is reduced to solving a certain class of nonlinear Schrodinger equations, whose most general solitonic solution is indeed available. Previously known solutions can be retrieved by taking appropriate limits in the parameters. The applicability of the method extends to systems with population imbalance and subject to external potential. In particular we show that fermionic zero-modes are robust against population imbalance.

Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity

For a large class of nonlinear Schrodinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity, we prove the existence of nontrivial finite energy traveling waves moving with speed c in any space dimension N?3. Our results are valid as well for the Gross-Pitaevskii equation and for NLS with cubic-quintic nonlinearity.

The existence of solutions for the problem in the optical lattice with nonlocal nonlinearity

We address the impact of nonlocality in the physical features exhibited by solitons supported by Kerr-type nonlinear media with an imprinted optical lattice. We discuss the solitons solution for a class of nonlinear Schrodinger equations in the optical lattice with nonlocal nonlinearity. We also show via a uniform priori estimate that existence and uniqueness of the global solution for the initial problem.

Blow-Up and Attractor of Solution for Problems of Nonlinear Schrodinger Equations

In this paper, the authors study the blow-up of solution for a class of nonlinear Schrodinger equation for some initial boundary problem. On the other hand, the authors give out some analyses and that new conclusion by Eigen-function method. In last section, the authors check the nonlinear parameter for light rule power by using of parameter method to get ground state and excite state correspond case, and discuss the global attractor of some fraction order case, and combine numerical test. To illustrate this physics meaning in dimension d = 1, 2 case. So, by numerable solution to give out these wave expression.

Nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential

This paper discusses a class of nonlinear Schrodinger equations with combined power-type nonlinearities and harmonic potential. By constructing a variational problem the potential well method is applied. The structure of the potential well and the properties of the depth function are given. The invariance of some sets for the problem is shown. It is proven that, if the initial data are in the potential well or out of it, the solutions will lie in the potential well or lie out of it, respectively. By the convexity method, the sharp condition of the global well-posedness is given.

Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials

We study the existence of positive solutions for a class of nonlinear Schrodinger equations of the type -epsilon(2)Delta u + Vu = u(p) in R-N, where N >= 3, p > 1 is subcritical and V is a nonnegative continuous potential. Amongst other results, we prove that if V has a positive local minimum, and N/N-2 0 around the local minimum point of V. The novelty is that no restriction is imposed on the rate of decay of V. In particular, we cover the case where V is compactly supported.

Solitons and spectral renormalization methods in nonlinear optics

Localized wave solutions, often referred to as solitary waves or solitons, are important classes of solutions in nonlinear optics. In optical communications, weakly nonlinear, quasi-monochromatic waves satisfy the “classical” and the “dispersion-managed” nonlocal nonlinear Schrodinger equations, both of which have localized pulses as special solutions. Recent research has shown that mode-locked lasers are also described by similar equations. These systems are variants of the classical nonlinear Schrodinger equation, appropriately modified to include terms which model gain, loss and spectral filtering that are present in the laser cavity. To study their remarkable properties, a computational method is introduced to find localized waves in nonlinear optical systems governed by these equations.

Bifurcations and the stability of the surface envelope solitons for a finite-depth fluid

The dynamics of the quasi-monochromatic surface gravitational waves in a finite-depth fluid is studied for the case where the product of the wavenumber by the depth of the fluid is close to the critical value kcrh ≈ 1.363. Within the framework of the Hamiltonian formalism, the general nonlinear Schrodinger equation is derived. In contrast to the classical nonlinear Schrodinger equation, this equation involves the gradient terms to the four-wave interaction, as well as the six-wave interaction. This equation is used to analyze the modulation instability of the monochromatic waves, as well as the bifurcations of the soliton solutions and their stability. It is shown that the solitons are stable and unstable to finite perturbations for focusing and defocusing nonlinearities, respectively.

Initial-boundary value problem of one class of nonlinear Schrödinger equations described in molecular crystals

In this paper, we study the initial-boundary value problem of one class of nonlinear Schrodinger equations described in molecular crystals. Furthermore, the existence of the global solution is obtained by means of interpolation inequality and a priori estimation.