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Zhenya Yan

Publications and source records attributed to Zhenya Yan.

At least 55 records · Page 3Linked to original sources
arXiv2021

Higher-dimensional soliton generation, stability and excitations of the PT-symmetric nonlinear Schrödinger equations

We study a class of physically intriguing PT-symmetric generalized Scarf-II (GS-II) potentials, which can support exact solitons in one- and multi-dimensional nonlinear Schrödinger equation. In the 1D and multi-D settings, we find that a properly adjusted localization parameter may support fully real energy spectra. Also, continuous families of fundamental and higher-order solitons are produced. The fundamental states are shown to be stable, while the higher-order ones, including 1D multimodal solitons, 2D solitons, and 3D light bullets, are unstable. Further, we find that the stable solitons can always propagate, in a robust form, remaining trapped in slowly moving potential wells of the GS-II type, which opens the way for manipulations of optical solitons. Solitons may also be transformed into stable forms by means of adibatic variation of potential parameters. Finally, an alternative type of n-dimensional PT-symmetric GS-II potentials is reported too. These results will be useful to further explore the higher-dimensional PT-symmetric solitons and to design the relative physical experiments.

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arXiv2021

Deep learning neural networks for the third-order nonlinear Schrodinger equation: Solitons, breathers, and rogue waves

The third-order nonlinear Schrodinger equation (alias the Hirota equation) is investigated via deep leaning neural networks, which describes the strongly dispersive ion-acoustic wave in plasma and the wave propagation of ultrashort light pulses in optical fibers, as well as broader-banded waves on deep water. In this paper, we use the physics-informed neural networks (PINNs) deep learning method to explore the data-driven solutions (e.g., soliton, breather, and rogue waves) of the Hirota equation when the two types of the unperturbated and unperturbated (a 2% noise) training data are considered. Moreover, we use the PINNs deep learning to study the data-driven discovery of parameters appearing in the Hirota equation with the aid of solitons.

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arXiv2021

Data-driven peakon and periodic peakon travelling wave solutions of some nonlinear dispersive equations via deep learning

In the field of mathematical physics, there exist many physically interesting nonlinear dispersive equations with peakon solutions, which are solitary waves with discontinuous first-order derivative at the wave peak. In this paper, we apply the multi-layer physics-informed neural networks (PINNs) deep learning to successfully study the data-driven peakon and periodic peakon solutions of some well-known nonlinear dispersion equations with initial-boundary value conditions such as the Camassa-Holm (CH) equation, Degasperis-Procesi equation, modified CH equation with cubic nonlinearity, Novikov equation with cubic nonlinearity, mCH-Novikov equation, b-family equation with quartic nonlinearity, generalized modified CH equation with quintic nonlinearity, and etc. These results will be useful to further study the peakon solutions and corresponding experimental design of nonlinear dispersive equations.

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arXiv2020

Higher-order vector Peregrine solitons and asymptotic estimates for the multi-component nonlinear Schrödinger equations

We first report the first- and higher-order vector Peregrine solitons (alias rational rogue waves) for the any multi-component NLS equations based on the loop group theory, an explicit (n + 1)-multiple eigenvalue of a characteristic polynomial of degree (n + 1) related to the condition of Benjamin-Feir instability, and inverse functions. Particularly, these vector rational rogue waves are parity-time symmetric for some parameter constraints. A systematic and effective approach is proposed to study the asymptotic behaviors of these vector rogue waves such that the decompositions of rogue waves are related to the so-called governing polynomials, which pave a powerful way in the study of vector rogue wave structures of the multi-component integrable systems. The vector rogue waves with maximal amplitudes can be determined via the parameter vectors, which is interesting and useful in the multi-component physical systems.

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arXiv2020

Parity-time-symmetric vector rational rogue wave solutions in any n-component nonlinear Schrödinger models

The extreme events are investigated for an $n$-component nonlinear Schrödinger ($n$-NLS) system in the focusing Kerr-like nonlinear media, which appears in many physical fields. We report and discuss the novel multi-parametric families of vector rational rogue wave (RW) solutions featuring the parity-time (PT) symmetry, which are characterized by non-identical boundary conditions for the components, and consistent with the degeneracy of $n$ branches of Benjamin-Feir instability. Explicit examples of PT-symmetric vector RWs are presented. Some parameter constraints can make some components generate the RWs with high amplitudes due to many-body resonant interactions.Effect of a non-integrable deformation of the model on the excitation of vector RWs is also discussed. These results will be useful to design the RW experiments in multi-component physical systems.

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arXiv2020

Rogue wave formation and interactions in the defocusing nonlinear Schrödinger equation with external potentials

The defocusing nonlinear Schrödinger (NLS) equation has no the modulational instability, and was not found to possess the rogue wave (RW) phenomenon up to now. In this paper, we firstly investigate some novel nonlinear wave structures in the defocusing NLS equation with real-valued time-dependent and time-independent potentials such that the stable new RWs and W-shaped solitons are found, respectively. Moreover, the interactions of two or three RWs are explored such that the RWs with higher amplitudes are generated in the defocusing NLS equation with real-valued time-dependent potentials. Finally, we study the defocusing NLS equation with complex PT -symmetric potentials such that some RWs and W-shaped solitons are also found. These novel results will be useful to design the related physical experiments to generate the RW phenomena and W-shaped solitons in the case of defocusing nonlinear interactions, and to apply them in the related fields of nonlinear or even linear sciences.

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arXiv2020

Data-driven rogue waves and parameter discovery in the defocusing NLS equation with a potential using the PINN deep learning

The physics-informed neural networks (PINNs) can be used to deep learn the nonlinear partial differential equations and other types of physical models. In this paper, we use the multi-layer PINN deep learning method to study the data-driven rogue wave solutions of the defocusing nonlinear Schrödinger (NLS) equation with the time-dependent potential by considering several initial conditions such as the rogue wave, Jacobi elliptic cosine function, two-Gaussian function, or three-hyperbolic-secant function, and periodic boundary conditions. Moreover, the multi-layer PINN algorithm can also be used to learn the parameter in the defocusing NLS equation with the time-dependent potential under the sense of the rogue wave solution. These results will be useful to further discuss the rogue wave solutions of the defocusing NLS equation with a potential in the study of deep learning neural networks.

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arXiv2020

Solving forward and inverse problems of the logarithmic nonlinear Schrodinger equation with PT-symmetric harmonic potential via deep learning

In this paper, we investigate the logarithmic nonlinear Schrödinger (LNLS) equation with the parity-time (PT)-symmetric harmonic potential, which is an important physical model in many fields such as nuclear physics, quantum optics, magma transport phenomena, and effective quantum gravity. Three types of initial value conditions and periodic boundary conditions are chosen to solve the LNLS equation with PT -symmetric harmonic potential via the physics-informed neural networks (PINNs) deep learning method, and these obtained results are compared with ones deduced from the Fourier spectral method. Moreover, we also investigate the effectiveness of the PINNs deep learning for the LNLS equation with PT symmetric potential by choosing the distinct space widths or distinct optimized steps. Finally, we use the PINNs deep learning method to effectively tackle the data-driven discovery of the LNLS equation with PT -symmetric harmonic potential such that the coefficients of dispersion and nonlinear terms or the amplitudes of PT -symmetric harmonic potential can be approximately found.

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arXiv2019

Attraction centers and PT-symmetric delta-functional dipoles in critical and supercritical self-focusing media

We introduce a model based on the one-dimensional nonlinear Schroedinger equation (NLSE) with the critical (quintic) or supercritical self-focusing nonlinearity. We demonstrate that a family of solitons, which are unstable in this setting against collapse, is stabilized by pinning to an attractive defect, that may also include a parity-time (PT)-symmetric gain-loss component. The model can be realized as a planar waveguide in optics, and in a super-Tonks-Girardeau bosonic gas. For the attractive defect with the delta-functional profile, a full family of the pinned solitons is found in an exact form. In the absence of the gain-loss term, the solitons' stability is investigated analytically too, by means of the Vakhitov-Kolokolov criterion; in the presence of the PT-balanced gain and loss, the stability is explored by means of numerical methods. In particular, the entire family of pinned solitons is stable in the quintic medium if the gain-loss term is absent. A stability region for the pinned solitons persists in the model with an arbitrarily high power of the self-focusing nonlinearity. A weak gain-loss component gives rise to alternations of stability and instability in the system's parameter plane. Those solitons which are unstable under the action of the supercritical self-attraction are destroyed by the collapse. If the self-attraction-driven instability is weak and the gain-loss term is present, unstable solitons spontaneously transform into localized breathers. The same outcome may be caused by a combination of the critical nonlinearity with the gain and loss. Instability of the solitons is also possible when the PT-symmetric gain-loss term is added to the subcritical nonlinearity. The system with self-repulsive nonlinearity is briefly considered too, producing completely stable families of pinned localized states.

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arXiv2019

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.

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arXiv2019

Inverse scattering transforms for the focusing and defocusing mKdV equations with nonzero boundary conditions

We explore systematically a rigorous theory of the inverse scattering transforms with matrix Riemann-Hilbert problems for both focusing and defocusing modified Korteweg-de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity. Using a suitable uniformization variable, the direct and inverse scattering problems are proposed on a complex plane instead of a two-sheeted Riemann surface. For the direct scattering problem, the analyticities, symmetries and asymptotic behaviors of the Jost solutions and scattering matrix, and discrete spectra are established. The inverse problems are formulated and solved with the aid of the Riemann-Hilbert problems, and the reconstruction formulae, trace formulae, and theta conditions are also found. In particular, we present the general solutions for the focusing mKdV equation with NZBCs and both simple and double poles, and for the defocusing mKdV equation with NZBCs and simple poles. Finally, some representative reflectionless potentials are in detail studied to illustrate distinct wave structures for both focusing and defocusing mKdV equations with NZBCs.

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arXiv2018

Inverse scattering transforms and N-double-pole solutions for the derivative NLS equation with zero/non-zero boundary conditions

We systematically report a rigorous theory of the inverse scattering transforms (ISTs) for the derivative nonlinear Schrodinger (DNLS) equation with both zero boundary condition (ZBC)/non-zero boundary conditions (NZBCs) at infinity and double poles of analytical scattering coefficients. The scattering theories for both ZBC and NZBCs are addressed. The direct problem establishes the analyticity, symmetries and asymptotic behavior of the Jost solutions and scattering matrix, and properties of discrete spectra. The inverse problems are formulated and solved with the aid of the matrix Riemann-Hilbert problems, and the reconstruction formulae, trace formulae and theta conditions are also posed. In particular, the IST with NZBCs at infinity is proposed by a suitable uniformization variable, which allows the scattering problem to be solved on a standard complex plane instead of a two-sheeted Riemann surface. The reflectionless potentials with double poles for the ZBC and NZBCs are both carried out explicitly by means of determinants. Some representative semi-rational bright-bright soliton, dark-bright soliton, and breather-breather solutions are examined in detail. These results will be useful to further explore and apply the related nonlinear wave phenomena.

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arXiv2018

Impact of near-PT symmetry on exciting solitons and interactions based on a complex Ginzburg-Landau model

We present and theoretically report the influence of a class of near-parity-time-(PT-) symmetric potentials with spectral filtering parameter $α_2$ and nonlinear gain-loss coefficient $β_2$ on solitons in the complex Ginzburg-Landau (CGL) equation. The potentials do not admit entirely-real linear spectra any more due to the existence of coefficients $α_2$ or $β_2$. However, we find that most stable exact solitons can exist in the second quadrant of the $(α_2, β_2)$ space, including on the corresponding axes. More intriguingly, the centrosymmetric two points in the $(α_2, β_2)$ space possess imaginary-axis (longitudinal-axis) symmetric linear-stability spectra. Furthermore, an unstable nonlinear mode can be excited to another stable nonlinear mode by the adiabatic change of $α_2$ and $β_2$. Other fascinating properties associated with the exact solitons are also examined in detail, such as the interactions and energy flux. These results are useful for the related experimental designs and applications.

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arXiv2017

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

We use two families of parameters $\{(ε_{x_j}, ε_{t_j})\,|\,ε_{x_j,t_j}=\pm1,\, j=1,2,...,n\}$ to first introduce a unified novel two-family-parameter system (simply called ${\mathcal Q}^{(n)}_{ε_{x_{\vec{n}}},ε_{t_{\vec{n}}}}$ system), connecting integrable local, nonlocal, novel mixed-local-nonlocal, and other nonlocal vector nonlinear Schrödinger (VNLS) equations. The ${\mathcal Q}^{(n)}_{ε_{x_{\vec{n}}}, ε_{t_{\vec{n}}}}$ system with $(ε_{x_j}, ε_{t_j})=(\pm 1, 1),\, j=1,2,...,n$ 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 $\{(ε_{x_j}, ε_{t_j}) | ε_{x_j} = e^{iθ_{x_j}}, ε_{t_j} = e^{iθ_{t_j}},\, θ_{x_j}, θ_{t_j}\in [0, 2π),\, j=1,2,...,n\}$ 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.

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arXiv2017

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

We introduce a new unified two-parameter $\{(ε_x, ε_t)\,|ε_{x,t}=\pm1\}$ wave model (simply called ${\mathcal Q}_{ε_x,ε_t}^{(n)}$ model), connecting integrable local and nonlocal vector nonlinear Schrödinger equations. The two-parameter $(ε_x, ε_t)$ family also brings insight into a one-to-one connection between four points $(ε_x, ε_t)$ (or complex numbers $ε_x+iε_t$) with $\{{\mathcal I}, {\mathcal P}, {\mathcal T}, {\mathcal PT}\}$ symmetries for the first time. The ${\mathcal Q}_{ε_x,ε_t}^{(n)}$ model with $(ε_x, ε_t)=(\pm 1, 1)$ 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 ${\mathcal Q}_{ε_x,ε_t}^{(n)}$ model including bright and dark solitons, periodic wave solutions, and multi-rogue wave solutions.

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arXiv2017

Solitons and their stability in the nonlocal nonlinear Schroedinger equation with PT-symmetric potentials

We report localized nonlinear modes of the self-focusing and defocusing nonlocal nonlinear Schroedinger equation with the generalized PT-symmetric Scarf-II, Rosen-Morse, and periodic potentials. Parameter regions are presented for broken and unbroken PT-symmetric phases of linear bounded states and the linear stability of the obtained solitons. Moreover, we numerically explore the dynamical behaviors of solitons and find stable solitons for some given parameters.

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arXiv2017

Initial-boundary value problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on a finite interval

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 can be expressed in terms of the solution of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Furthermore, the relevant jump matrices with explicit (x, t)-dependence of the matrix RH problem can be explicitly found via three spectral functions {s(k), S(k), S_L(k)} arising from the initial data and the Dirichlet-Neumann boundary conditions at x=0 and x=L, respectively. The global relation is also found to deduce two distinct but equivalent types of representations (i.e., one via the large $k$ of asymptotics of the eigenfunctions and another one in terms of the Gel'fand-Levitan-Marchenko (GLM) approach) for the Dirichlet and Neumann boundary value problems. In particular, the formulae for IBV problems on the finite interval can reduce to ones on a half-line as the length $L$ of the interval approaches to infinity. Moreover, we also present the linearizable boundary conditions for the GLM representations.

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