Inverse Scattering Transform and Solitons for Square Matrix Nonlinear Schrödinger Equations

The inverse scattering transform (IST) is developed for a class of matrix nonlinear Schrödinger‐type systems whose reductions include two equations that model certain hyperfine spin spinor Bose–Einstein condensates, and two novel equations that were recently shown to be integrable, and that have applications in nonlinear optics and four‐component fermionic condensates. In addition, the general behavior of the soliton solutions for all four reductions is analyzed in detail, and some novel solutions are presented.     

Soliton Interactions for the Three‐Coupled Discrete Nonlinear Schrödinger Equations in the Alpha Helical Proteins

Three‐coupled discrete nonlinear Schrödinger equations, which describe the dynamics of the three hydrogen bonding spines in the alpha helical proteins with the interspine coupling at the discrete level, are investigated. Binary Bell polynomials are applied to construct the bilinear forms and bilinear Bäcklund transformation of those equations. Propagation characteristics and interactions of the bound‐state solitons are discussed. Bound states of two and three bright solitons arise when all of them propagate in parallel. Elastic interaction between the bound‐state solitons and one bright soliton is given. Increase of the dipole‐dipole interaction energy can lead to the increase of the soliton velocity, that is, the one‐interaction period becomes shorter.     

Instabilities of Multihump Vector Solitons in Coupled Nonlinear Schrödinger Equations

Spectral stability of multihump vector solitons in the Hamiltonian system of coupled nonlinear Schrödinger (NLS) equations is investigated both analytically and numerically. Using the closure theorem for the negative index of the linearized Hamiltonian, we classify all possible bifurcations of unstable eigenvalues in the systems of coupled NLS equations with cubic and saturable nonlinearities. We also determine the eigenvalue spectrum numerically by the shooting method. In case of cubic nonlinearities, all multihump vector solitons in the nonintegrable model are found to be linearly unstable. In case of saturable nonlinearities, stable multihump vector solitons are found in certain parameter regions, and some errors in the literature are corrected.     

Ground State and Geometrically Distinct Solitons of Discrete Nonlinear Schrödinger Equations with Saturable Nonlinearities

We study the discrete nonlinear equation where (the spectrum of L) and is asymptotically linear as for all . We obtain the existence of ground state solitons and the existence of infinitely many pairs of geometrically distinct solitons of this equation. Our method is based on the generalized Nehari manifold method developed recently by Szulkin and Weth. To the best of our knowledge, this technique has not been used for discrete equations with saturable nonlinearities.     

Coupled Nonlinear Schrödinger Equations with a Gauge Potential: Existence and Blowup

We study solutions of the Cauchy problem for nonlinear Schrödinger system in with nonlinear coupling and linear coupling modeling synthetic magnetic field in spin‐orbit coupled Bose–Einstein condensates. Three main results are presented: a proof of the local existence, a proof of the sufficient condition for the blowup result in finite time for some solutions, and the persistence of the nonlinear dynamics in the limit where the spin‐orbit coupling converges to zero.     

Probabilistic Methods for Discrete Nonlinear Schrödinger Equations

We show that the thermodynamics of the focusing cubic discrete nonlinear Schrödinger equation are exactly solvable in dimension 3 and higher. A number of explicit formulas are derived. © 2012 Wiley Periodicals, Inc.     

Designable Integrability of the Variable Coefficient Nonlinear Schrödinger Equations

The designable integrability (DI) [ 51 ] of the variable coefficient nonlinear Schrödinger equations (VCNLSEs) is first introduced by construction of an explicit transformation, which maps VCNLSE to the usual nonlinear Schrödinger equations (NLSEs). One novel feature of VCNLSE with DI is that its coefficients can be designed artificially and analytically by using transformation. A special example between nonautonomous NLSEs and NLSEs is given here. Further, the optical super‐lattice potentials (or periodic potentials) and multiwell potentials are designed, which are two kinds of important potential in Bose–Einstein condensation and nonlinear optical systems. There are two interesting features of the soliton of the VCNLSEs indicated by the analytic and exact formula. Specifically, its profile is variable and its trajectory is not a straight line when it evolves with time t.     

Classification of Solitary Wave Bifurcations in Generalized Nonlinear Schrödinger Equations

Bifurcations of solitary waves are classified for the generalized nonlinear Schrödinger equations with arbitrary nonlinearities and external potentials in arbitrary spatial dimensions. Analytical conditions are derived for three major types of solitary wave bifurcations, namely, saddle‐node, pitchfork, and transcritical bifurcations. Shapes of power diagrams near these bifurcations are also obtained. It is shown that for pitchfork and transcritical bifurcations, their power diagrams look differently from their familiar solution‐bifurcation diagrams. Numerical examples for these three types of bifurcations are given as well. Of these numerical examples, one shows a transcritical bifurcation, which is the first report of transcritical bifurcations in the generalized nonlinear Schrödinger equations. Another shows a power loop phenomenon which contains several saddle‐node bifurcations, and a third example shows double pitchfork bifurcations. These numerical examples are in good agreement with the analytical results.     

On the Integrability of the Poisson Driven Stochastic Nonlinear Schrödinger Equations

We consider the Cauchy problem for the dissipative nonlinear Schrödinger equations driven by a Poisson noise, namely (1) where γ_n > 0 and 0 < t₁ < ? < t_n < ? are certain sequences of random numbers and is the deterministic loss coefficient. This perturbation incorporates the possibility of sudden changes in the field that occur randomly. If Γ= 0, we prove that the resulting equation can be piece‐wise related to the unperturbed NLS equation and show how to solve the initial value problem. We also determine a complete set of conserved quantities. When Γ≠ 0 the equation is nonintegrable. Nevertheless, we determine the random evolution of physically relevant quantities like the field’s Energy E(t) ≡∫dx|u|²(t, x) and momentum. By considering a joint z‐Laplace transform we obtain the mean Energy decay. A naturally related quantity is the “half‐life”, or the time before the Energy degrades below a given value E₁. We show that the mean of this random quantity satisfies an integral equation and solve it by Laplace transformation. In particular cases we also determine the complete probability distribution of Energy and half life.     

High‐Order Solutions and Generalized Darboux Transformations of Derivative Nonlinear Schrödinger Equations

By means of certain limit technique, two kinds of generalized Darboux transformations are constructed for the derivative nonlinear Schrödinger equations (DNLS). These transformations are shown to lead to two solution formulas for DNLS in terms of determinants. As applications, several different types of high‐order solutions are calculated for this equation.     

Initial‐Boundary Value Problems for the Coupled Nonlinear Schrödinger Equation on the Half‐Line

Initial‐boundary value problems for the coupled nonlinear Schrödinger equation on the half‐line are investigated via the Fokas method. It is shown that the solution can be expressed in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k‐plane, whose jump matrix is defined in terms of the matrix spectral functions and that depend on the initial data and all boundary values, respectively. If there exist spectral functions satisfying the global relation, it can be proved that the function defined by the above Riemann–Hilbert problem solves the coupled nonlinear Schrödinger equation and agrees with the prescribed initial and boundary values. The most challenging problem in the implementation of this method is to characterize the unknown boundary values that appear in the spectral function . For a particular class of boundary conditions so‐called linearizable boundary conditions, it is possible to compute the spectral function in terms of and given boundary conditions by using the algebraic manipulation of the global relation. For the general case of boundary conditions, an effective characterization of the unknown boundary values can be obtained by employing perturbation expansion.     

Discretization of Coupled Nonlinear Schrödinger Equations

The soliton solutions for discrete coupled nonlinear Schrödinger equations are investigated by using bilinear formalism. Pfaffian expressions of the N -soliton solutions of dark–dark and bright–bright types are explicitly given for the defocusing–defocusing and focusing–focusing cases, respectively.     

The Self-Focusing Singularity in the Nonlinear Schrödinger Equation

Under certain circumstances, solutions of the cylindrically symmetric nonlinear Schrödinger equation collapse to a singularity in a finite time. An asymptotic series for the solution near the singularity is derived here. At leading order, the central amplitude of the spike grows like[(log Δt)/Δt]^1/2, where Δt is the time remaining to the appearance of the singularity.     

Concentration on curves for nonlinear Schrödinger Equations

We consider the problem where p > 1, ε > 0 is a small parameter, and V is a uniformly positive, smooth potential. Let Γ be a closed curve, nondegenerate geodesic relative to the weighted arc length ∫_Γ V^σ, where σ = (p + 1)/(p ? 1) ? 1/2. We prove the existence of a solution u_? concentrating along the whole of Γ, exponentially small in ε at any positive distance from it, provided that ε is small and away from certain critical numbers. In particular, this establishes the validity of a conjecture raised in 3 in the two‐dimensional case. © 2006 Wiley Periodicals, Inc.     

On the Global Behavior of Solutions of a Coupled System of Nonlinear Schrödinger Equation

We mainly study a system of two coupled nonlinear Schrödinger equations where one equation includes gain and the other one includes losses. This model constitutes a generalization of the model of pulse propagation in birefringent optical fibers. We aim in this study at partially answering a question of some authors in [1]: “Is the H¹‐norm of the solution globally bounded in the Manakov case, when ?” We found that in the Manakov case, and when , the solution stays in , and also that the H¹‐norm of the solution cannot blow up in finite time. In the Manakov case, an estimate of the total energy is provided, which is different from that has been given in [1]. These results are corroborated by numerical results that have been obtained with a finite element solver well adapted for that purpose.     

耦合非线性Schrödinger方程组的Neumann问题

该文考虑一类耦合椭圆型非线性Schr\"{o}dinger方程组的Neumann问题极小能量解（基态解）的存在性和集中性质. 主要研究极小能量解的尖点, 即最大值点的位置. 利用 Lin Tai-Chia 和 Wei Juncheng 研究 Dirichlet 问题的方法, 该文首先得到了相应Neumann问题的极小能量解的存在性. 当相当于Planck常数的小参数趋于零时, 该文证明了极小能量解的尖点向定义区域的边界靠近, 并且能量集中在这些尖点处. 另外, 方程组解的两个分支解相互吸引或排斥时, 它们的尖点也相互吸引或排斥.