arXiv subjects

Jianke Yang

Publications and source records attributed to Jianke Yang.

At least 37 records · Page 2Linked to original sources

arXiv2021

Universal rogue wave patterns associated with the Yablonskii-Vorob'ev polynomial hierarchy

We show that universal rogue wave patterns exist in integrable systems. These rogue patterns comprise fundamental rogue waves arranged in shapes such as triangle, pentagon and heptagon, with a possible lower-order rogue wave at the center. These patterns appear when one of the internal parameters in bilinear expressions of rogue waves gets large. Analytically, these patterns are determined by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy through a linear transformation. Thus, the induced rogue patterns in the space-time plane are simply the root structures of Yablonskii-Vorob'ev polynomials under actions such as dilation, rotation, stretch, shear and translation. As examples, these universal rogue patterns are explicitly determined and graphically illustrated for the generalized derivative nonlinear Schrödinger equations, the Boussinesq equation, and the Manakov system. Similarities and differences between these rogue patterns and those reported earlier in the nonlinear Schrödinger equation are discussed.

nlin.SI↗

arXiv2021

Analytical construction of soliton families in one- and two-dimensional nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials

The existence of soliton families in non-parity-time-symmetric complex potentials remains poorly understood, especially in two spatial dimensions. In this article, we analytically investigate the bifurcation of soliton families from linear modes in one- and two-dimensional nonlinear Schrödinger equations with localized Wadati-type non-parity-time-symmetric complex potentials. By utilizing the conservation law of the underlying non-Hamiltonian wave system, we convert the complex soliton equation into a new real system. For this new real system, we perturbatively construct a continuous family of low-amplitude solitons bifurcating from a linear eigenmode to all orders of the small soliton amplitude. Hence, the emergence of soliton families in these non-parity-time-symmetric complex potentials is analytically explained. We also compare these analytically constructed soliton solutions with high-accuracy numerical solutions in both one and two dimensions, and the asymptotic accuracy of these perturbation solutions is confirmed.

nlin.PS↗

arXiv2021

General rogue waves in the three-wave resonant interaction systems

General rogue waves in (1+1)-dimensional three-wave resonant interaction systems are derived by the bilinear method. These solutions are divided into three families, which correspond to a simple root, two simple roots and a double root of a certain quartic equation arising from the dimension reduction respectively. It is shown that while the first family of solutions associated with a simple root exist for all signs of the nonlinear coefficients in the three-wave interaction equations, the other two families of solutions associated with two simple roots and a double root can only exist in the so-called soliton-exchange case, where the nonlinear coefficients have certain signs. Many of these rogue wave solutions, such as those associated with two simple roots, and higher-order solutions associated with a simple root, are new solutions which have not been reported before. Technically, our bilinear derivation of rogue waves for the double-root case is achieved by a generalization to the previous dimension reduction procedure in the bilinear method, and this generalized procedure allows us to treat roots of arbitrary multiplicities. Dynamics of the derived rogue waves is also examined, and new rogue-wave patterns are presented. Connection between these bilinear rogue waves and those derived earlier by Darboux transformation is also explained.

nlin.SI↗

arXiv2021

Rogue wave patterns in the nonlinear Schrödinger equation

Rogue wave patterns in the nonlinear Schrödinger equation are analytically studied. It is shown that when an internal parameter in the rogue waves (which controls the shape of initial weak perturbations to the uniform background) is large, these waves would exhibit clear geometric structures, which are formed by Peregrine waves in shapes such as triangle, pentagon, heptagon and nonagon, with a possible lower-order rogue wave at its center. These rogue patterns are analytically determined by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy, and their orientations are controlled by the phase of the large parameter. It is also shown that when multiple internal parameters in the rogue waves are large but satisfy certain constraints, similar rogue patterns would still hold. Comparison between true rogue patterns and our analytical predictions shows excellent agreement.

nlin.SI↗

arXiv2020

Weakly nonlinear topological gap solitons in Su-Schrieffer-Heeger photonic lattices

We study both theoretically and experimentally the effect of nonlinearity on topologically protected linear interface modes in a photonic Su-Schrieffer-Heeger (SSH) lattice. It is shown that under either focusing or defocusing nonlinearity, this linear topological mode of the SSH lattice turns into a family of topological gap solitons. These solitons are stable. However, they exhibit only a low amplitude and power and are thus weakly nonlinear, even when the bandgap of the SSH lattice is wide. As a consequence, if the initial beam has modest or high power, it will either delocalize, or evolve into a soliton not belonging to the family of topological gap solitons. These theoretical predictions are observed in our experiments with optically induced SSH-type photorefractive lattices.

physics.optics↗

arXiv2020

Universal patterns of rogue waves

Rogue wave patterns in the nonlinear Schrödinger (NLS) equation and the derivative NLS equation are analytically studied. It is shown that when the free parameters in the analytical expressions of these rogue waves are large, these waves would exhibit the same patterns, comprising fundamental rogue waves forming clear geometric structures such as triangle, pentagon, heptagon and nonagon, with a possible lower-order rogue wave at its center. These rogue patterns are analytically determined by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy, and their orientations are controlled by the phase of the large free parameter. This connection of rogue wave patterns to the root structures of the Yablonskii-Vorob'ev polynomial hierarchy goes beyond the NLS and derivative NLS equations, and it gives rise to universal rogue wave patterns in integrable systems.

nlin.PS↗

arXiv2020

New families of non-parity-time-symmetric complex potentials with all-real spectra

New families of non-parity-time-symmetric complex potentials with all-real spectra are derived by the supersymmetry method and the pseudo-Hermiticity method. With the supersymmetry method, we find families of non-parity-time-symmetric complex partner potentials which share the same spectrum as base potentials with known real spectra, such as the (complex) Wadati potentials. Different from previous supersymmetry derivations of potentials with real spectra, our derivation does not utilize discrete eigenmodes of base potentials. As a result, our partner potentials feature explicit analytical expressions which contain free functions. With the pseudo-Hermiticity method, we derive a new class of non-parity-time-symmetric complex potentials with free functions and constants, whose eigenvalues appear as conjugate pairs. This eigenvalue symmetry forces the spectrum to be all-real for a wide range of choices of these functions and constants in the potential. Tuning these free functions and constants, phase transition can also be induced, where conjugate pairs of complex eigenvalues emerge in the spectrum.

math-ph↗

arXiv2019

Rogue waves in the generalized derivative nonlinear Schrodinger equations

General rogue waves are derived for the generalized derivative nonlinear Schrodinger (GDNLS) equations by a bilinear Kadomtsev-Petviashvili (KP) reduction method. These GDNLS equations contain the Kaup-Newell equation, the Chen-Lee-Liu equation and the Gerdjikov-Ivanov equation as special cases. In this bilinear framework, it is shown that rogue waves to all members of these equations are expressed by the same bilinear solution. Compared to previous bilinear KP reduction methods for rogue waves in other integrable equations, an important improvement in our current KP reduction procedure is a new parameterization of internal parameters in rogue waves. Under this new parameterization, the rogue wave expressions through elementary Schur polynomials are much simpler. In addition, the rogue wave with the highest peak amplitude at each order can be obtained by setting all those internal parameters to zero, and this maximum peak amplitude at order $N$ turns out to be $2N+1$ times the background amplitude, independent of the individual GDNLS equation and the background wavenumber. It is also reported that these GDNLS equations can be decomposed into two different bilinear systems which require different KP reductions, but the resulting rogue waves remain the same. Dynamics of rogue waves in the GDNLS equations is also analyzed. It is shown that the wavenumber of the constant background strongly affects the orientation and duration of the rogue wave. In addition, some new rogue patterns are presented.

nlin.SI↗

arXiv2019

General rogue waves in the Boussinesq equation

We derive general rogue wave solutions of arbitrary orders in the Boussinesq equation by the bilinear Kadomtsev-Petviashvili (KP) reduction method. These rogue solutions are given as Gram determinants with $2N-2$ free irreducible real parameters, where $N$ is the order of the rogue wave. Tuning these free parameters, rogue waves of various patterns are obtained, many of which have not been seen before. Compared to rogue waves in other integrable equations, a new feature of rogue waves in the Boussinesq equation is that the rogue wave of maximum amplitude at each order is generally asymmetric in space. On the technical aspect, our contribution to the bilinear KP-reduction method for rogue waves is a new judicious choice of differential operators in the procedure, which drastically simplifies the dimension reduction calculation as well as the analytical expressions of rogue wave solutions.

nlin.SI↗

arXiv2019

On general rogue waves in the parity-time-symmetric nonlinear Schrodinger equation

This article addresses the question of general rogue-wave solutions in the nonlocal parity-time-symmetric nonlinear Schrodinger equation. By generalizing the previous bilinear Kadomtsev-Petviashvili reduction method, large classes of rogue waves are derived as Gram determinants with Schur polynomial elements. It is shown that these rogue waves contain previously reported ones as special cases. More importantly, they contain many new rogue wave families. It is conjectured that the rogue waves derived in this article are all rogue-wave solutions in the parity-time-symmetric nonlinear Schrodinger equation.

nlin.SI↗

arXiv2019

Symmetry breaking with opposite stability between bifurcated asymmetric solitons in parity-time-symmetric potentials

We report a new type of symmetry-breaking bifurcation of solitons in optical systems with parity-time-symmetric potentials. In this bifurcation, the two bifurcated branches of asymmetric solitons exhibit opposite stability, which contrasts all previous symmetry-breaking bifurcations in conservative and non-conservative systems. We show that this novel symmetry-breaking bifurcation can be exploited to achieve unidirectional propagation of high-intensity light beams in parity-time-symmetric potentials.

nlin.PS↗

arXiv2018

Construction of non-PT-symmetric complex potentials with all-real spectra

We review recent work on the generalization of PT symmetry. We show that, in addition to PT-symmetric complex potentials, there are also large classes of non-PT-symmetric complex potentials which also feature all-real spectra. In addition, some classes of these non-PT-symmetric potentials allow phase transitions which do or do not go through exceptional points. These non-PT-symmetric potentials are constructed by a variety of methods, such as the symmetry and supersymmetry methods and the soliton theory. A generalization of PT symmetry in multi-dimensions is also reviewed.

math-ph↗

arXiv2018

A new nonlocal nonlinear Schroedinger equation and its soliton solutions

A new integrable nonlocal nonlinear Schroedinger (NLS) equation with clear physical motivations is proposed. This equation is obtained from a special reduction of the Manakov system, and it describes Manakov solutions whose two components are related by a parity symmetry. Since the Manakov system governs wave propagation in a wide variety of physical systems, this new nonlocal equation has clear physical meanings. Solitons and multi-solitons in this nonlocal equation are also investigated in the framework of Riemann-Hilbert formulations. Surprisingly, symmetry relations of discrete scattering data for this equation are found to be very complicated, where constraints between eigenvectors in the scattering data depend on the number and locations of the underlying discrete eigenvalues in a very complex manner. As a consequence, general $N$-solitons are difficult to obtain in the Riemann-Hilbert framework. However, one- and two-solitons are derived, and their dynamics investigated. It is found that two-solitons are generally not a nonlinear superposition of one-solitons, and they exhibit interesting dynamics such as meandering and sudden position shifts. As a generalization, other integrable and physically meaningful nonlocal equations are also proposed, which include NLS equations of reverse-time and reverse-space-time types as well as nonlocal Manakov equations of reverse-space, reverse-time and reverse-space-time types.

nlin.SI↗

arXiv2017

General $N$-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations

General $N$-solitons in three recently-proposed nonlocal nonlinear Schrödinger equations are presented. These nonlocal equations include the reverse-space, reverse-time, and reverse-space-time nonlinear Schrödinger equations, which are nonlocal reductions of the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. It is shown that general $N$-solitons in these different equations can be derived from the same Riemann-Hilbert solutions of the AKNS hierarchy, except that symmetry relations on the scattering data are different for these equations. This Riemann-Hilbert framework allows us to identify new types of solitons with novel eigenvalue configurations in the spectral plane. Dynamics of $N$-solitons in these equations is also explored. In all the three nonlocal equations, a generic feature of their solutions is repeated collapsing. In addition, multi-solitons can behave very differently from fundamental solitons and may not correspond to a nonlinear superposition of fundamental solitons.

nlin.SI↗

arXiv2017

General rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger equation

Rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger (NLS) equation are studied by Darboux transformation. Three types of rogue waves are derived, and their explicit expressions in terms of Schur polynomials are presented. These rogue waves show a much wider variety than those in the local NLS equation. For instance, the polynomial degrees of their denominators can be not only $n(n+1)$, but also $n(n-1)+1$ and $n^2$, where $n$ is an arbitrary positive integer. Dynamics of these rogue waves is also examined. It is shown that these rogue waves can be bounded for all space and time, or develop collapsing singularities, depending on their types as well as values of their free parameters. In addition, the solution dynamics exhibits rich patterns, most of which have no counterparts in the local NLS equation.

nlin.SI↗

arXiv2017

New classes of non-parity-time-symmetric optical potentials with all-real spectra and exceptional-point-free phase transition

Paraxial linear propagation of light in an optical waveguide with material gain and loss is governed by a Schrödinger equation with a complex potential. Properties of parity-time-symmetric complex potentials have been heavily studied before. In this article, new classes of non-parity-time-symmetric complex potentials featuring conjugate-pair eigenvalue symmetry in its spectrum are constructed by operator symmetry methods. Due to this eigenvalue symmetry, it is shown that the spectrum of these complex potentials is often all-real. Under parameter tuning in these potentials, phase transition can also occur, where pairs of complex eigenvalues appear in the spectrum. A peculiar feature of the phase transition here is that, the complex eigenvalues may bifurcate out from an interior continuous eigenvalue inside the continuous spectrum, in which case a phase transition takes place without going through an exceptional point. In one spatial dimension, this class of non-parity-time-symmetric complex potentials is of the form $V(x)=h'(x)-h^2(x)$, where $h(x)$ is an arbitrary parity-time-symmetric complex function. These potentials in two spatial dimensions are also derived. Diffraction patterns in these complex potentials are further examined, and unidirectional propagation behaviors are demonstrated.

physics.optics↗

arXiv2017

Transformations between nonlocal and local integrable equations

Recently, a number of nonlocal integrable equations, such as the PT-symmetric nonlinear Schrodinger (NLS) equation and PT-symmetric Davey-Stewartson equations, were proposed and studied. Here we show that many of such nonlocal integrable equations can be converted to local integrable equations through simple variable transformations. Examples include these nonlocal NLS and Davey-Stewartson equations, a nonlocal derivative NLS equation, the reverse space-time complex modified Korteweg-de Vries (CMKdV) equation, and many others. These transformations not only establish immediately the integrability of these nonlocal equations, but also allow us to construct their analytical solutions from solutions of the local equations. These transformations can also be used to derive new nonlocal integrable equations. As applications of these transformations, we use them to derive rogue wave solutions for the partially PT-symmetric Davey-Stewartson equations and the nonlocal derivative NLS equation. In addition, we use them to derive multi-soliton and quasi-periodic solutions in the reverse space-time CMKdV equation. Furthermore, we use them to construct many new nonlocal integrable equations such as nonlocal short pulse equations, nonlocal nonlinear diffusion equations, and nonlocal Sasa-Satsuma equations.

nlin.PS↗

arXiv2016

Nonlinear behaviors of parity-time-symmetric lasers

We propose a time-dependent partial differential equation model to investigate the dynamical behavior of the parity-time (PT) symmetric laser during the nonlinear stage of its operation. This model incorporates physical effects such as the refractive index distribution, dispersion, material loss, nonlinear gain saturation and self-phase modulation. We show that when the loss is weak, multiple stable steady states and time-periodic states of light exist above the lasing threshold, rendering the laser multi-mode. However, when the loss is strong, only a single stable steady state of broken PT symmetry exists for a wide range of the gain amplitude, rendering the laser single-mode. These results reveal the important role the loss plays in maintaining the single-mode operation of PT lasers.

physics.optics↗