Stable embedded solitons

Stable embedded solitons are discovered in the generalized third-order nonlinear Schr?dinger equation. When this equation can be reduced to a perturbed complex modified Korteweg-de Vries equation, we developed a soliton perturbation theory which shows that a continuous family of sech-shaped embedded solitons exist and are nonlinearly stable. These analytical results are confirmed by our numerical simulations. These results establish that, contrary to previous beliefs, embedded solitons can be robust despite being in resonance with the linear spectrum.     

Power controlled soliton stability and steering in lattices with saturable nonlinearity

Dynamical properties of discrete solitons in nonlinear Schr?dinger lattices with saturable nonlinearity are studied in the framework of the one-dimensional discrete Vinetskii-Kukhtarev model. Two stationary strongly localized modes, centered on site (A) and between two neighboring sites (B), are obtained. The associated Peierls-Nabarro potential is bounded and has multiple zeros indicating strong implications on the stability and dynamics of the localized modes. Besides a stable propagation of mode A, a stable propagation of mode B is also possible. The enhanced ability of the large power solitons to move across the lattice is pointed out and numerically verified.     

Modulated spin waves and robust quasi-solitons in classical Heisenberg rings

We investigate the dynamical behavior of finite rings of classical spin vectors interacting via nearest-neighbor isotropic exchange in an external magnetic field. Our approach is to utilize the solutions of a continuum version of the discrete spin equations of motion (EOM) which we derive by assuming continuous modulations of spin wave solutions of the EOM for discrete spins. This continuum EOM reduces to the Landau-Lifshitz equation in a particular limiting regime. The usefulness of the continuum EOM is demonstrated by the fact that the time-evolved numerical solutions of the discrete spin EOM closely track the corresponding time-evolved solutions of the continuum equation. It is of special interest that our continuum EOM possesses soliton solutions, and we find that these characteristics are also exhibited by the corresponding solutions of the discrete EOM. The robustness of solitons is demonstrated by considering cases where initial states are truncated versions of soliton states and by numerical simulations of the discrete EOM equations when the spins are coupled to a heat bath at finite temperatures.     

Multicolor lattice solitons

We report on the existence of multicolor solitons supported by periodic lattices made from quadratic nonlinear media. Such lattice solitons bridge the gap between continuous solitons in uniform media and discrete solitons in strongly localized systems and exhibit a wealth of new features. We discovered that, in contrast to uniform media, multipeaked lattice solitons are stable. Thus they open new opportunities for all-optical switching based on soliton packets.     

Internal modes of discrete solitons near the anti-continuum limit of the dNLS equation

Discrete solitons of the discrete nonlinear Schrödinger (dNLS) equation are compactly supported in the anti-continuum limit of the zero coupling between lattice sites. Eigenvalues of the linearization of the dNLS equation at the discrete soliton determine its spectral stability. Small eigenvalues bifurcating from the zero eigenvalue near the anti-continuum limit were characterized earlier for this model. Here we analyze the resolvent operator and prove that it is bounded in the neighborhood of the continuous spectrum if the discrete soliton is simply connected in the anti-continuum limit. This result rules out the existence of internal modes (neutrally stable eigenvalues of the discrete spectrum) near the anti-continuum limit.     

Backward-wave propagation and discrete solitons in a left-handed electrical lattice

We study experimentally, analytically and numerically the backward-wave propagation, and formation of discrete bright and dark solitons in a nonlinear electrical lattice. We observe experimentally that a focusing (defocusing) effect occurs above (below) a certain carrier frequency threshold, and backward-propagating bright (dark) discrete solitons are formed. We develop a discrete model emulating the relevant circuit and benchmark its linear properties against the experimental dispersion relation. Using a perturbation method, we derive a nonlinear Schrödinger equation, that predicts accurately the carrier frequency threshold. Finally, we use numerical simulations to corroborate our findings and monitor the space-time evolution of the discrete solitons.     

Observation of in-band lattice solitons

We report the first experimental and theoretical demonstrations of in-band (or embedded) lattice solitons. Such solitons appear in trains, and their propagation constants reside inside the first Bloch band of a square lattice, different from all previously observed solitons. We show that these solitons bifurcate from Bloch modes at the interior high-symmetry X points within the first band, where normal and anomalous diffractions coexist along two orthogonal directions. At high powers, the in-band soliton can move into the first band gap and turn into a gap soliton.     

空间调制作用下Bessel型光晶格中物质波孤立子的稳定性

利用变分法和数值计算方法研究了空间调制作用下Bessel型光晶格中玻色-爱因斯坦凝聚体系中孤立子的稳定性, 给出了存在随空间非周期变化的线性Bessel型光晶格和非线性光晶格(原子之间非线性相互作用的空间调制)时, 各种参数组合下涡旋和非涡旋孤立子的稳定性条件. 首先, 利用圆对称的高斯型试探波函数得出描述体系稳定性参数满足的Euler-Lagrange方程和变分法分析体系稳定性所需要的有效作用势能的表达式. 然后, 根据有效作用势能是否具有局域最小值判断体系是否具有稳定状态, 得出体系具有稳定状态时参数所满足的条件. 最后, 利用有限差分法求解Gross-Pitaevskii方程验证变分法结果的正确性, 所得结果和变分法结果一致. 关键词： Bessel型光晶格 非线性光晶格 孤立子 稳定性     

Fundamental solitons in discrete lattices with a delayed nonlinear response

The formation of unstaggered localized modes in dynamical lattices can be supported by the interplay of discreteness and nonlinearity with a finite relaxation time. In rapidly responding nonlinear media, on-site discrete solitons are stable, and their broad intersite counterparts are marginally stable, featuring a virtually vanishing real instability eigenvalue. The solitons become unstable in the case of the slowly relaxing nonlinearity. The character of the instability alters with the increase of the delay time, which leads to a change in the dynamics of unstable discrete solitons. They form robust localized breathers in rapidly relaxing media, and decay into oscillatory diffractive pattern in the lattices with a slow nonlinear response. Marginally stable solitons can freely move across the lattice.     

Two-dimensional matter-wave solitons and vortices in competing cubic-quintic nonlinear lattices

The nonlinear lattice — a new and nonlinear class of periodic potentials — was recently introduced to generate various nonlinear localized modes. Several attempts failed to stabilize two-dimensional (2D) solitons against their intrinsic critical collapse in Kerr media. Here, we provide a possibility for supporting 2D matter-wave solitons and vortices in an extended setting — the cubic and quintic model — by introducing another nonlinear lattice whose period is controllable and can be different from its cubic counterpart, to its quintic nonlinearity, therefore making a fully “nonlinear quasi-crystal”.A variational approximation based on Gaussian ansatz is developed for the fundamental solitons and in particular, their stability exactly follows the inverted Vakhitov–Kolokolov stability criterion, whereas the vortex solitons are only studied by means of numerical methods. Stability regions for two types of localized mode — the fundamental and vortex solitons — are provided. A noteworthy feature of the localized solutions is that the vortex solitons are stable only when the period of the quintic nonlinear lattice is the same as the cubic one or when the quintic nonlinearity is constant, while the stable fundamental solitons can be created under looser conditions. Our physical setting (cubic-quintic model) is in the framework of the Gross–Pitaevskii equation or nonlinear Schrödinger equation, the predicted localized modes thus may be implemented in Bose–Einstein condensates and nonlinear optical media with tunable cubic and quintic nonlinearities.     

Spatial solitons in an optical lattice with varying diffraction and spatially inhomogeneous nonlinearity

In this paper, we present the (1+1)-dimensional inhomogeneous nonlinear Schrödinger (NLS) equation that describes the propagation of optical waves in nonlinear optical systems exhibiting optical lattice, inhomogeneous nonlinearity and varying diffraction at the same time. A series of interesting properties of spatial solitons are found from the numerical calculations, such as the stable propagation in the a nonperiodic optical lattice induced by periodic diffraction variations and periodic nonlinearity variations. Finally, the interaction of neighboring spatial solitons in a nonperiodic optical lattice is discussed, and the results reveal that two spatial solitons can propagate periodically and separately in the optical lattice without interaction.     

Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity

We study the discrete nonlinear Schrödinger lattice model with the onsite nonlinearity of the general form, |u|^2σu. We systematically verify the conditions for the existence and stability of discrete solitons in the one-dimensional version of the model predicted by means of the variational approximation (VA), and demonstrate the following: monostability of fundamental solitons (FSs) in the case of the weak nonlinearity, 2σ+1<3.68; bistability, in a finite range of values of the soliton’s power, for 3.68<2σ+1<5; and the presence of a threshold (minimum norm of the FS), for 2σ+1≥5. We also perform systematic numerical simulations to study higher-order solitons in the same general model, i.e., bound states of the FSs. While all in-phase bound states are unstable, stability regions are identified for antisymmetric double solitons and their triple counterparts. These numerical findings are supplemented by an analytical treatment of the stability problem, which allows quantitively accurate predictions for the stability features of such multipulses. When these waveforms are found to be unstable, we show, by means of direct simulations, that they self-trap into a persistent lattice breather, or relax into a stable FS, or sometimes decay completely.     

Semidiscrete composite solitons in arrays of quadratically nonlinear waveguides

We demonstrate that an array of discrete waveguides on a slab substrate, both featuring chi2 nonlinearity, supports stable solitons composed of discrete and continuous components. Two classes of fundamental composite soliton are identified: ones consisting of a discrete fundamental-frequency (FF) component in the waveguide array, coupled to a continuous second-harmonic (SH) component in the slab waveguide, and solitons with an inverted FF/SH structure. Twisted bound states of the fundamental solitons are found, too. In contrast with the usual systems, the intersite-centered fundamental solitons and bound states with the twisted continuous components are stable over almost the entire domain of their existence.     

Stable soliton complexes in two-dimensional photonic lattices

We show that two-dimensional photonic Kerr nonlinear lattices can support stable soliton complexes composed of several solitons packed together with appropriately engineered phases. This may open up new prospects for encoding pixellike images made of robust discrete or lattice solitons.     

The soliton properties of dipole domains in superlattices

The formation and propagation of dipole domains in superlattices are studied both by the modified discrete drift model and by the nonlinear schroedinger equation,the spatiotemporal distribution of the electric field and electron density are presented.The numerical results are compared with the soliton solutions of the nonlinear Schroedinger equation and analysed.It is shown that the numerical solutions agree with the soliton solutions of the nonlinear Schroedinger equation.The dipole electric-field domains in semiconductor superlattices have the properties of solitons.     

Integrable Hierarchy Covering the Lattice Burgers Equation in Fluid Mechanics:N-fold Darboux Transformation and Conservation Laws

Burgers-type equations can describe some phenomena in fluids,plasmas,gas dynamics,traffic,etc.In this paper,an integrable hierarchy covering the lattice Burgers equation is derived from a discrete spectral problem.N-fold Darboux transformation(DT) and conservation laws for the lattice Burgers equation are constructed based on its Lax pair.N-soliton solutions in the form of Vandermonde-like determinant are derived via the resulting DT with symbolic computation,structures of which are shown graphically.Coexistence of the elastic-inelastic interaction among the three solitons is firstly reported for the lattice Burgers equation,even if the similar phenomenon for certern continuous systems is known.Results in this paper might be helpful for understanding some ecological problems describing the evolution of competing species and the propagation of nonlinear waves in fluids.     

Integrable Hierarchy Covering the Lattice Burgers Equation in Fluid Mechanics: N-fold Darboux Transformation and Conservation Laws

Burgers-type equations can describe some phenomena in fluids, plasmas, gas dynamics, traffic, etc. In this paper, an integrable hierarchy covering the lattice Burgers equation is derived from a discrete spectral problem. N-fold Darboux transformation (DT) and conservation laws for the lattice Burgers equation are constructed based on its Lax pair. N-soliton solutions in the form of Vandermonde-like determinant are derived via the resulting DT with symbolic computation, structures of which are shown graphically. Coexistence of the elastic-inelastic interaction among the three solitons is firstly reported for the lattice Burgers equation, even if the similar phenomenon for certern continuous systems is known. Results in this paper might be helpful for understanding some ecological problems describing the evolution of competing species and the propagation of nonlinear waves in fluids.     

Dark-bright discrete solitons: A numerical study of existence, stability and dynamics

In the present work, we numerically explore the existence and stability properties of different types of configurations of dark-bright solitons, dark-bright soliton pairs and pairs of dark-bright and dark solitons in discrete settings, starting from the anti-continuum limit. We find that while single discrete dark-bright solitons have similar stability properties to discrete dark solitons, their pairs may only be stable if the bright components are in phase and are always unstable if the bright components are out of phase. Pairs of dark-bright solitons with dark ones have similar stability properties as individual dark or dark-bright ones. Lastly, we consider collisions between dark-bright solitons and between a dark-bright one and a dark one. Especially in the latter and in the regime where the underlying lattice structure matters, we find a wide range of potential dynamical outcomes depending on the initial soliton speed.     

Dipole and quadrupole solitons in optically-induced two-dimensional defocusing photonic lattices

Dipole and quadrupole solitons in a two-dimensional optically induced defocusing photonic lattice are theoretically predicted and experimentally observed. It is shown that in-phase nearest-neighbor and out-of-phase next-nearest-neighbor dipoles exist and can be stable in the intermediate intensity regime. There are also different types of dipoles that are always unstable. In-phase nearest-neighbor quadrupoles are also numerically obtained, and may also be linearly stable. Out-of-phase, nearest-neighbor quadrupoles are found to be typically unstable. These numerical results are found to be aligned with the main predictions obtained analytically in the discrete nonlinear Schrödinger model. Finally, experimental results are presented for both dipole and quadrupole structures, indicating that self-trapping of such structures in the defocusing lattice can be realized for the length of the nonlinear crystal (10 mm).