Localized Analytical Solutions and Parameters Analysis in the Nonlinear Dispersive Gross-Pitaevskii Mean-Field GP (m,n) Model with Space-Modulated Nonlinearity and Potential

The novel nonlinear dispersive Gross-Pitaevskii (GP) mean-field model with the space-modulated nonlinearity and potential (called GP(m, n) equation) is investigated in this paper. By using self-similar transformations and some powerful methods, we obtain some families of novel envelope compacton-like solutions spikon-like solutions to the GP(n, n) (n>1) equation. These solutions possess abundant localized structures because of infinite choices of the self-similar function X(x). In particular, we choose X(x) as the Jacobi amplitude function am(x,k) and the combination of linear and trigonometric functions of space x so that the novel localized structures of the GP(2,2) equation are illustrated, which are much different from the usual compacton and spikon solutions reported. Moreover, it is shown that GP(m,1) equation with linear dispersion also admits the compacton-like solutions for the case 0<m<1 and spikon-like solutions for the case m<0.Comment: 16 pages, 9 figure

An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4x4 Lax pair on the half-line

We investigate the initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii (GP) equations with a 4x4 Lax pair on the half-line. The solution of this system can be obtained in terms of the solution of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. The relevant jump matrices of the RH problem can be explicitly found using the two spectral functions s(k) and S(k), which can be defined by the initial data, the Dirichlet-Neumann boundary data at x=0. The global relation is established between the two dependent spectral functions. The general mappings between Dirichlet and Neumann boundary values are analyzed in terms of the global relation.Comment: 30 pages, 3 figure

An initial-boundary value problem of the general three-component nonlinear Schrodinger equation with a 4x4 Lax pair on a finite interval

We investigate the initial-boundary value problem for the general three-component nonlinear Schrodinger (gtc-NLS) equation with a 4x4 Lax pair on a finite interval by extending the Fokas unified approach. The solutions of the gtc-NLS equation can be expressed in terms of the solutions of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Moreover, the relevant jump matrices of the RH problem can be explicitly found via the three spectral functions arising from the initial data, the Dirichlet-Neumann boundary data. The global relation is also established to deduce two distinct but equivalent types of representations (i.e., one by using the large k of asymptotics of the eigenfunctions and another one in terms of the Gelfand-Levitan-Marchenko (GLM) method) for the Dirichlet and Neumann boundary value problems. Moreover, the relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval approaches to infinity. Finally, we also give the linearizable boundary conditions for the GLM representation.Comment: 58 pages, 3 figure

Integrable PT{\mathcal PT}-symmetric local and nonlocal vector nonlinear Schr\"odinger equations: a unified two-parameter model

We introduce a new unified two-parameter $\{(\epsilon_x, \epsilon_t)\,|\epsilon_{x,t}=\pm1\}$ wave model (simply called ${\mathcal Q}_{\epsilon_x,\epsilon_t}^{(n)}$ model), connecting integrable local and nonlocal vector nonlinear Schr\"odinger equations. The two-parameter $(\epsilon_x, \epsilon_t)$ family also brings insight into a one-to-one connection between four points $(\epsilon_x, \epsilon_t)$ (or complex numbers $\epsilon_x+i\epsilon_t$) with $\{{\mathcal I}, {\mathcal P}, {\mathcal T}, {\mathcal PT}\}$ symmetries for the first time. The ${\mathcal Q}_{\epsilon_x,\epsilon_t}^{(n)}$ model with $(\epsilon_x, \epsilon_t)=(\pm 1, 1)$ is shown to possess a Lax pair and infinite number of conservation laws, and to be ${\mathcal PT}$ symmetric. Moreover, the Hamiltonians with self-induced potentials are shown to be ${\mathcal PT}$ symmetric only for ${\mathcal Q}_{-1,-1}^{(n)}$ model and to be ${\mathcal T}$ symmetric only for ${\mathcal Q}_{+1,-1}^{(n)}$ model. The multi-linear form and some self-similar solutions are also given for the ${\mathcal Q}_{\epsilon_x,\epsilon_t}^{(n)}$ model including bright and dark solitons, periodic wave solutions, and multi-rogue wave solutions.Comment: 6 pages, 1 figure, submitted on Jan. 25, 2015 (corrected version

A novel hierarchy of two-family-parameter equations: Local, nonlocal, and mixed-local-nonlocal vector nonlinear Schrodinger equations

We use two families of parameters $\{(\epsilon_{x_j}, \epsilon_{t_j})\,|\,\epsilon_{x_j,t_j}=\pm1,\, j=1,2,...,n\}$ to first introduce a unified novel two-family-parameter system (simply called ${\mathcal Q}^{(n)}_{\epsilon_{x_{\vec{n}}},\epsilon_{t_{\vec{n}}}}$ system), connecting integrable local, nonlocal, novel mixed-local-nonlocal, and other nonlocal vector nonlinear Schr\"odinger (VNLS) equations. The ${\mathcal Q}^{(n)}_{\epsilon_{x_{\vec{n}}}, \epsilon_{t_{\vec{n}}}}$ system with $(\epsilon_{x_j}, \epsilon_{t_j})=(\pm 1, 1),\, j=1,2,...,n$ is shown to possess Lax pairs and infinite number of conservation laws. Moreover, we also analyze the ${\mathcal PT}$ symmetry of the Hamiltonians with self-induced potentials. The multi-linear forms and some symmetry reductions are also studied. In fact, the used two families of parameters can also be extended to the general case $\{(\epsilon_{x_j}, \epsilon_{t_j}) | \epsilon_{x_j} = e^{i\theta_{x_j}}, \epsilon_{t_j} = e^{i\theta_{t_j}},\, \theta_{x_j}, \theta_{t_j}\in [0, 2\pi),\, j=1,2,...,n\}$ to generate more types of nonlinear equations. The two-family-parameter idea used in this paper can also be applied to other local nonlinear evolution equations such that novel integrable and non-integrable nonlocal and mixed-local-nonlocal systems can also be found.Comment: 8 page

Optical rogue waves in the generalized inhomogeneous higher-order nonlinear Schrodinger equation with modulating coefficients

The higher-order dispersive and nonlinear effects (alias {\it the perturbation terms}) like the third-order dispersion, the self-steepening, and the self-frequency shift play important roles in the study of the ultra-short optical pulse propagation. We consider optical rogue wave solutions and interactions for the generalized higher-order nonlinear Schr\"odinger (NLS) equation with space- and time-modulated parameters. A proper transformation is presented to reduce the generalized higher-order NLS equation to the integrable Hirota equation with constant coefficients. This transformation allows us to relate certain class of exact solutions of the generalized higher-order NLS equation to the variety of solutions of the integrable Hirota equation. In particular, we illustrate the approach in terms of two lowest-order rational solutions of the Hirota equation as seeding functions to generate rogue wave solutions localized in time that have complicated evolution in space with or without the differential gain or loss term. We simply analyze the physical mechanisms of the obtained optical rogue waves on the basis of these constraints. Finally, The stability of the obtained rogue-wave solutions is addressed numerically. The obtained rogue wave solutions may raise the possibility of relative experiments and potential applications in nonlinear optics and other fields of nonlinear science as Bose-Einstein condensates and oceanComment: 12 pages, 10 figure

Stable parity-time-symmetric nonlinear modes and excitations in a derivative nonlinear Schrodinger equation

The effect of derivative nonlinearity and parity-time- (PT-) symmetric potentials on the wave propagation dynamics is investigated in the derivative nonlinear Schrodinger equation, where the physically interesting Scarff-II and hamonic-Hermite-Gaussian potentials are chosen. We study numerically the regions of unbroken/broken linear PT-symmetric phases and find some stable bright solitons of this model in a wide range of potential parameters even though the corresponding linear PT-symmetric phases are broken. The semi-elastic interactions between exact bright solitons and exotic incident waves are illustrated such that we find that exact nonlinear modes almost keep their shapes after interactions even if the exotic incident waves have evidently been changed. Moreover, we exert the adiabatic switching on PT-symmetric potential parameters such that a stable nonlinear mode with the unbroken linear PT-symmetric phase can be excited to another stable nonlinear mode belonging to the broken linear PT-symmetric phase.Comment: 11 pages, 11 figure

Solitonic dynamics and excitations of the nonlinear Schrodinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials

Solitons are of the important significant in many fields of nonlinear science such as nonlinear optics, Bose-Einstein condensates, plamas physics, biology, fluid mechanics, and etc.. The stable solitons have been captured not only theoretically and experimentally in both linear and nonlinear Schrodinger (NLS) equations in the presence of non-Hermitian potentials since the concept of the parity-time (PT)-symmetry was introduced in 1998. In this paper, we present novel bright solitons of the NLS equation with third-order dispersion in some complex PT-symmetric potentials (e.g., physically relevant PT-symmetric Scarff-II-like and harmonic-Gaussian potentials). We find stable nonlinear modes even if the respective linear PT-symmetric phases are broken. Moreover, we also use the adiabatic changes of the control parameters to excite the initial modes related to exact solitons to reach stable nonlinear modes. The elastic interactions of two solitons are exhibited in the third-order NLS equation with PT-symmetric potentials. Our results predict the dynamical phenomena of soliton equations in the presence of third-order dispersion and PT-symmetric potentials arising in nonlinear fiber optics and other physically relevant fields.Comment: 11 pages, 8 figure

A unified inverse scattering transform and soliton solutions of the nonlocal modified KdV equation with non-zero boundary conditions

We present a rigorous theory of a unified and simple inverse scattering transform (IST) for both focusing and defocusing real nonlocal (reverse-space-time) modified Korteweg-de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity. The IST problems for the nonlocal equations with NZBCs are more complicated then ones for the local equations with NZBCs. The suitable uniformization variable is introduced in order to make the direct and inverse problems be established on a complex plane instead of a two-sheeted Riemann surface. The direct scattering problem establishes the analyticity, symmetries, and asymptotic behaviors of Jost solutions and scattering matrix, and properties of discrete spectra. The inverse problem is formulated and solved by means of a matrix-valued Riemann-Hilbert problem. The reconstruction formula, trace formulae, and theta conditions are obtained. Finally, the dynamical behaviors of solitons for four different cases for the reflectionless potentials for both focusing and defocusing nonlocal mKdV equations with NZBCs are analyzed in detail.Comment: 21 pages, 4 figure

Matter-wave solutions in the Bose-Einstein condensates with the harmonic and Gaussian potentials

We study exact solutions of the quasi-one-dimensional Gross-Pitaevskii (GP) equation with the (space, time)-modulated potential and nonlinearity and the time-dependent gain or loss term in Bose-Einstein condensates. In particular, based on the similarity transformation, we report several families of exact solutions of the GP equation in the combination of the harmonic and Gaussian potentials, in which some physically relevant solutions are described. The stability of the obtained matter-wave solutions is addressed numerically such that some stable solutions are found. Moreover, we also analyze the parameter regimes for the stable solutions. These results may raise the possibility of relative experiments and potential applications.Comment: 10 pages, 10 figure