arXiv subjects

Jingsong He

Publications and source records attributed to Jingsong He.

At least 19 recordsLinked to original sources

arXiv2026

The variable-length stem structures in three-soliton resonance of the Kadomtsev-Petviashvili II equation

The stem structure is a localized feature that arises during high-order soliton interactions, connecting the vertices of two V-shaped waveforms. The interaction of resonant 3-solitons is accompanied by soliton reconnection phenomena, characterized by the disappearance and reconnection of stem structures. This paper investigates variable-length stem structures in resonant 3-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation, focusing on both 2-resonant and 3-resonant 3-soliton cases. Depending on the phase shift tends to plus/minus infinity, different types of resonances are identified, including strong resonance, weak resonance, and mixed (strong-weak) resonance. We derive and analyze the asymptotic forms and explicit expressions for the soliton arm trajectories, velocities, as well as the endpoints, length, and amplitude of the stem structures. A detailed comparison is made between the similarities and differences of the stem structures in the 2-resonant and 3-resonant solitons. In addition, we provide a comprehensive and rigorous analysis of both the asymptotic behavior and the structural properties of the stems.

math-ph↗

arXiv2026

Localized stem structures in quasi-resonant solutions of the Kadomtsev-Petviashvili equation

When the phase shift of X-shaped solutions before and after interaction is finite but approaches infinity, the vertices of the two V-shaped structures become separated due to the phase shift and are connected by a localized structure, which is referred to as the stem structure. This special type of elastic collision is known as a quasi-resonant collision. This study investigates quasi-resonant solutions and the associated localized stem structures in the context of the KPII and KPI equations. For the KPII equation, we classify quasi-resonant 2-solitons into weakly and strongly types, depending on whether the phase approaches $-\infty$ or $+\infty$. We analyze their asymptotic forms to detail the trajectories, amplitudes, velocities, and lengths of their stem structures. These results of quasi-resonant 2-solitons are used to to provide analytical descriptions of interesting patterns of the water waves observed on shallow water surface. Similarly, for the KPI equation, we construct quasi-resonant breather-soliton solutions and classify them into weakly and strongly types, based on the behavior of their internal parameters. We compare the similarities and differences between the stem structures in the quasi-resonant soliton and the quasi-resonant breather-soliton. Additionally, we provide a comprehensive and rigorous analysis of their asymptotic forms and stem structures. Our results indicate that the resonant solution, i.e. resonant breather-soliton of the KPI and soliton for the KPII, represents the limiting case of the quasi-resonant solution as phase approaches $\infty$.

nlin.PS↗

arXiv2026

Multiple double-valley and single-valley dark solitons in the complex modified Korteweg-de Vries equation: shape-preserving collisions and shape-altering collisions

The shape-preserving and shape-altering collisions of dark solitons are investigated in the complex modified Korteweg-de Vries equation. The obtained dark soliton solutions are classified into two distinct types, referred to as type-I and type-II dark solitons, which exhibit fundamentally different structural and dynamical characteristics. A single type-I dark soliton is symmetric about its center and admits three distinct valley profiles, namely single-valley, double-valley, and flat-bottom structures, whereas a single type-II dark soliton only supports a single-valley profile. These two types of dark solitons differ in their phase behaviors as the spatial variable varies from $-\infty$ to $+\infty$, as well as in their velocity--amplitude relations. For multiple pure type-I dark solitons, collisions are shape-preserving; however, a nontrivial collective effect is revealed in which modifying the parameters of one soliton can induce changes in the profiles and amplitudes of the other solitons, even though all collisions remain elastic in nature. In contrast, multiple pure type-II dark solitons behave independently, undergoing only phase shifts without any modification of their shapes or amplitudes. When type-I and type-II dark solitons coexist, their interactions lead to genuine shape-altering collisions, where the valley structures and amplitudes of type-I dark solitons are modified, while type-II solitons remain unaffected except for phase shifts. Asymptotic analysis further shows that the influence of type-II dark solitons on type-I dark solitons is confined to the pre-collision stage and disappears after the interaction.

nlin.SI↗

arXiv2025

Localized stem structures in soliton reconnection of the asymmetric Nizhnik-Novikov-Veselov system

The reconnection processes of 3-solitons with 2-resonance can produce distinct local structures that initially connect two pairs of V-shaped branches, then disappear, and later re-emerge as new forms. We call such local structures as stem structures. In this paper, we investigate the variable-length stem structures during the soliton reconnection of the asymmetric Nizhnik-Novikov-Veselov system. We consider two scenarios: weak 2-resonances (i.e., $a_{12}=a_{13}=0,\,0<a_{23}<+\infty$) and strong 2-resonances (i.e., $a_{12}=a_{13}=+\infty,\,0<a_{23}<+\infty$). We determine the asymptotic forms of the four arms and their corresponding stem structures using two-variable asymptotic analysis method which is involved simultaneously with one space variable $y$ (or $x$) and one temporal variable $t$. Different from known studies, our findings reveal that the asymptotic forms of the arms $S_2$ and $S_3$ differ by a phase shift as $t\to\pm\infty$. Building on these asymptotic forms, we perform a detailed analysis of the trajectories, amplitudes, and velocities of the soliton arms and stem structures. Subsequently, we discuss the localization of the stem structures, focusing on their endpoints, lengths, and extreme points in both weak and strong 2-resonance scenarios.

math-ph↗

arXiv2025

Self-gravitating nonlinear Dirac fields

We consider the self-gravitating Dirac field with a scalar fermion self-interaction term. For strong enough attractive fermion self-interaction, the maximum Arnowitt-Deser-Misner mass of soliton solutions consisting of two fermions can exceed the limit of noninteracting Dirac stars classically, and the "particle-like" solutions of the system can exhibit multiplicity, multiple valid solutions for a single set of parameters, that is inherent in the nonlinear Dirac field. We also find the mass-scale separation in our system similar to that discussed in the Einstein-Dirac-Higgs system. Interestingly, the system admits parity-violated solutions. The broken parity symmetry can be restored by increasing the central redshift of the solution.

gr-qc↗

arXiv2025

Soliton and traveling wave solutions in coupled one-dimensional condensates

Ultracold condensates provide a unique platform for exploring soliton physics. Motivated by the recent experiments realizing the sine-Gordon model in a split one-dimensional (1D) BEC, we demonstrate that this system naturally supports various density and phase solitons. We explore the physics using the bosonization technique, in which the phase and density are conjugate pairs, and determine its effective Language equation and the associated equation of motion. We show that in the presence of asymmetry between the two condensates, new solutions beyond those in the sine-Gordon model emerge. We calculate the traveling wave solutions and soliton solutions in this model and determine their corresponding energy densities analytically. Finally, we discuss the relevance of these solutions to the experiments and discuss their observations. This theory does not rely on the mechanism of quasi-particle excitation, which yields the Lee-Huang-Yang correction in higher dimensions, and is thus much more suitable to describe the physics in 1D systems. Since the physical models have already been realized in experiments, this work opens a new frontier for the realization of various soliton and periodic solutions using two coupled condensates.

cond-mat.quant-gas↗

arXiv2025

Internal dynamics and fission of pure-quartic soliton molecules

We address the weak interaction of a pair of well-separated pure-quartic solitons (PQSs), which are solutions to a generalized nonlinear Schrodinger equation (NLSE) with the quartic-only dispersion. An asymptotic technique is applied to derive equations for the slow evolution of the temporal separation and phase difference of the PQSs interacting through the overlapping of their exponentially decaying oscillating tails. Based on this approach, various stationary states of bound PQS (soliton molecules) with distinct phase differences are predicted. Their stability is addressed via the numerical calculation of the eigenvalue spectrum of small perturbations, showing instability of the bound states. A systematic numerical analysis demonstrates that the parameter space of the PQS bound states is organized as a self-similar fractal structure, composed of regions populated by robustly oscillating or splitting two-soliton states. The analytical method and results reported here can be extended for bound states of two or several weakly interacting modes in other conservative and dissipative systems.

physics.optics↗

arXiv2025

Long-time behavior of the reduced Maxwell-Bloch equations in the sharp-line limit

We study the Cauchy problem for the reduced Maxwell-Bloch equations with initial data for the electric field in weighted Sobolev spaces, assuming that all atoms initially reside in their ground state. Using the d-bar steepest descent method, we derive long-time asymptotic expansions of the solutions, including both the electric field and the components of the Bloch vector, within any fixed cone. In particular, we formulate the inverse scattering transform as a properly posed Riemann-Hilbert problem, avoiding singularities in the scattering data by modifying the time evolution of the reflection coefficient. Under assumptions that allow only soliton generation, the leading-order asymptotics are determined by solitons inside the cone, while soliton-radiation interactions appear in lower-order terms. These results extend the applicability of the nonlinear steepest descent method to integrable systems with singularities in the associated Lax pair.

math.AP↗

arXiv2024

Solutions of generalized constrained discrete KP hierarchy

Solutions of a generalized constrained discrete KP (gcdKP) hierarchy with constraint on Lax operator $L^k=(L^k)_{\geq m}+\sum_{i=1}^lq_iΔ^{-1}Λ^mr_i$, are invesitigated by Darboux transformations $T_D(f)=f^{[1]}\cdotΔ\cdot f^{-1}$ and $T_I(g)=(g^{[-1]})^{-1}\cdotΔ^{-1}\cdot g$. Due to this special constraint on Lax operator, it is showed that the generating functions $f$ and $g$ of the corresponding Darboux transformations, can only be chosen from (adjoint) wave functions or $(L^k)_{<m}=\sum_{i=1}^lq_iΔ^{-1}Λ^mr_i$. Then successive applications of Darboux transformations for gcdKP hierarchy are discussed. Finally based upon above, solutions of gcdKP hierarchy are obtained from $L^{\{0\}}=Λ$ by Darboux transformations.

nlin.SI↗

arXiv2024

Localized stem structures in quasi-resonant two-soliton solutions for the asymmetric Nizhnik-Novikov-Veselov system

Elastic collisions of solitons generally have a finite phase shift. When the phase shift has a finitely large value, the two vertices of the (2+1)-dimensional 2-soliton are significantly separated due to the phase shift, accompanied by the formation of a local structure connecting the two V-shaped solitons. We define this local structure as the stem structure. This study systematically investigates the localized stem structures between two solitons in the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov system. These stem structures, arising from quasi-resonant collisions between the solitons, exhibit distinct features of spatial locality and temporal invariance. We explore two scenarios: one characterized by weakly quasi-resonant collisions (i.e. $a_{12}\approx 0$), and the other by strongly quasi-resonant collisions (i.e. $a_{12}\approx +\infty$). Through mathematical analysis, we extract comprehensive insights into the trajectories, amplitudes, and velocities of the soliton arms. Furthermore, we discuss the characteristics of the stem structures, including their length and extreme points. Our findings shed new light on the interaction between solitons in the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov system.

nlin.SI↗

arXiv2024

Lifetime Characterization of Extreme Wave Localizations in Crossing Seas

Rogue waves (RWs) can form on the ocean surface due to quasi-four wave resonant interaction or superposition principle. Both mechanisms have been acutely studied. The first of the two is known as the nonlinear focusing mechanism and leads to an increased probability of rogue waves when wave conditions are favourable, i.e., when unidirectionality and high narrowband energy of the wave field are satisfied. This work delves into the dynamics of extreme wave focusing in crossing seas, revealing a distinct type of nonlinear RWs, characterized by a decisive longevity compared to those generated by the dispersive focusing mechanism. In fact, through fully nonlinear hydrodynamic numerical simulations, we show that the interactions between two crossing unidirectional wave beams can trigger fully localized and robust development of RWs. These coherent structures, characterized by a typical spectral broadening then spreading in the form of dual bimodality and recurrent wave group focusing, not only defy the weakening expectation of quasi-four wave resonant interaction in directionally spread wave fields, but also differ from classical focusing mechanisms already mentioned. This has been determined following a rigorous lifespan-based statistical analysis of extreme wave events in our fully nonlinear simulations. Utilizing the coupled nonlinear Schrödinger framework, we also show that such intrinsic focusing dynamics can also be captured by weakly nonlinear wave evolution equations. This opens new research avenues for further explorations of these complex and intriguing wave phenomena in hydrodynamics as well as other nonlinear and dispersive multi-wave systems.

physics.flu-dyn↗

arXiv2023

A solvable model for symmetry-breaking phase transitions

Analytically solvable models are benchmarks in studies of phase transitions and pattern-forming bifurcations. Such models are known for phase transitions of the second kind in uniform media, but not for localized states (solitons), as integrable equations which produce solitons do not admit intrinsic transitions in them. We introduce a solvable model for symmetry-breaking phase transitions of both the first and second kinds (alias sub- and supercritical bifurcations) for solitons pinned to a combined linear-nonlinear double-well potential, represented by a symmetric pair of delta-functions. Both self-focusing and defocusing signs of the nonlinearity are considered. In the former case, exact solutions are produced for symmetric and asymmetric solitons. The solutions explicitly demonstrate a switch between the symmetry-breaking transitions of the first and second kinds (i.e., sub- and supercritical bifurcations, respectively). In the self-defocusing model, the solution demonstrates the transition of the second kind which breaks antisymmetry of the first excited state.

nlin.PS↗

arXiv2022

Asymptotic dynamics of higher-order lumps in the Davey-Stewartson II equation

A family of higher-order rational lumps on non-zero constant background of Davey-Stewartson (DS) II equation are investigated. These solutions have multiple peaks whose heights and trajectories are approximately given by asymptotical analysis. It is found that the heights are time-dependent and for large time they approach the same constant height value of the first-order fundamental lump. The resulting trajectories are considered and it is found that the scattering angle can assume arbitrary values in the interval of $(\fracπ{2}, π)$ which is markedly distinct from the necessary orthogonal scattering for the higher-order lumps on zero background. Additionally, it is illustrated that the higher-order lumps containing multi-peaked $n$-lumps can be regarded as a nonlinear superposition of $n$ first-order ones as $|t|\rightarrow\infty$.

nlin.SI↗

arXiv2022

General higher-order breathers and rogue waves in the two-component long-wave--short-wave resonance-interaction model

General higher-order breather and rogue wave (RW) solutions to the two-component long wave--short wave resonance interaction (2-LSRI) model are derived via the bilinear Kadomtsev-Petviashvili hierarchy reduction method and are given in terms of determinants. Under particular parametric conditions, the breather solutions can reduce to homoclinic orbits, or a mixture of breathers and homoclinic orbits. There are three families of RW solutions, which correspond to a simple root, two simple roots, and a double root of an algebraic equation related to the dimension reduction procedure. The first family of RW solutions consists of $\frac{N(N+1)}{2}$ bounded fundamental RWs, the second family is composed of $\frac{N_1(N_1+1)}{2}$ bounded fundamental RWs coexisting with another $\frac{N_2(N_2+1)}{2}$ fundamental RWs of different bounded state ($N,N_1,N_2$ being positive integers), while the third one have ${[\widehat{N}_1^2+\widehat{N}_2^2-\widehat{N}_1(\widehat{N}_2-1)]}$ fundamental bounded RWs ($\widehat{N}_1,\widehat{N}_2$ being non-negative integers). The second family can be regarded as the superpositions of the first family, while the third family can be the degenerate case of the first family under particular parameter choices. These diverse RW patterns are illustrated graphically.

nlin.SI↗

arXiv2022

$L^2$ Sobolev space bijectivity of the scattering-inverse scattering transforms related to defocusing Ablowitz-Ladik systems

In this paper, we establish $L^2$-Sobolev space bijectivity of the inverse scattering transform related to the defocusing Ablowitz-Ladik system. On the one hand, in the direct problem, based on the spectral problem, we establish the reflection coefficient and the corespondent Riemann-Hilbert problem. And we also prove that if the potential belongs to $l^{2,k}$ space, then the reflection coefficient belongs to $H^k_θ(Σ)$. On the other hand, in the inverse problem, based on the Riemann-Hilbert problem, we obtain the corespondent reconstructed formula and recover potentials from reflection coefficients. And we also confirm that if reflection coefficients are in $H^k_θ(Σ)$, then we show that potentials also belong to $l^{2,k}$. This study also confirm that for the initial-valued problem of defocusing Ablowitz-Ladik equations, it the initial potential belongs to $l^{2,k}$ and satisfying $\parallel q\parallel_\infty<1$, then the solution for $t\ne0$ also belongs to $l^{2,k}$.

math.AP↗

arXiv2020

Rogue waves and lumps on the non-zero background in the PT -symmetric nonlocal Maccari system

In this paper, the PT -symmetric version of the Maccari system is introduced, which can be regarded as a two-dimensional generalization of the defocusing nonlocal nonlinear Schrodinger equation. Various exact solutions of the nonlocal Maccari system are obtained by means of the Hirota bilinear method, long-wave limit, and Kadomtsev-Petviashvili (KP) hierarchy method. Bilinear forms of the nonlocal Maccari system are derived for the first time. Simultaneously, a new nonlocal Davey-Stewartson-type equation is derived. Solutions for breathers and breathers on top of periodic line waves are obtained through the bilinear form of the nonlocal Maccari system. Hyperbolic line rogue-wave solutions and semi-rational ones, composed of hyperbolic line rogue wave and periodic line waves are also derived in the long-wave limit. The semi-rational solutions exhibit a unique dynamical behavior. Additionally, general line soliton solutions on constant background are generated by restricting different tau-functions of the KP hierarchy, combined with the Hirota bilinear method. These solutions exhibit elastic collisions, some of which have never been reported before in nonlocal systems. Additionally, the semi-rational solutions, namely (i) fusion of line solitons and lumps into line solitons, and (ii) fission of line solitons into lumps and line solitons, are put forward in terms of the KP hierarchy. These novel semi-rational solutions reduce to 2N-lump solutions of the nonlocal Maccari system with appropriate parameters. Finally, different characteristics of exact solutions for the nonlocal Maccari system are summarized. These new results enrich the structure of waves in nonlocal nonlinear systems, and help to understand new physical phenomena.

nlin.SI↗

arXiv2020

Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg-de Vries equation

Within the (2 + 1)-dimensional Korteweg-de Vries equation framework, new bilinear Backlund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation. By introducing an arbitrary function, a family of deformed soliton and deformed breather solutions are presented with the improved Hirotas bilinear method. Choosing the appropriate parameters, their interesting dynamic behaviors are shown in three-dimensional plots. Furthermore, novel rational solutions are generated by taking the limit of obtained solitons. Additionally, two dimensional [2D] rogue waves (localized in both space and time) on the soliton plane are presented, we refer to it as deformed 2D rogue waves. The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane, and its evolution process is analyzed in detail. The deformed 2D rogue wave solutions are constructed successfully, which are closely related to the arbitrary function. This new idea is also applicable to other nonlinear systems.

nlin.SI↗

arXiv2020

Reductions of the (4 + 1)-dimensional Fokas equation and their solutions

An integrable extension of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations is investigated in this paper.We will refer to this integrable extension as the (4+1)-dimensional Fokas equation. The determinant expressions of soliton, breather, rational, and semi-rational solutions of the (4 + 1)-dimensional Fokas equation are constructed based on the Hirota's bilinear method and the KP hierarchy reduction method. The complex dynamics of these new exact solutions are shown in both three-dimensional plots and two-dimensional contour plots. Interestingly, the patterns of obtained high-order lumps are similar to those of rogue waves in the (1 + 1)-dimensions by choosing different values of the free parameters of the model. Furthermore, three kinds of new semi-rational solutions are presented and the classification of lump fission and fusion processes is also discussed. Additionally, we give a new way to obtain rational and semi-rational solutions of (3 + 1)-dimensional KP equation by reducing the solutions of the (4 + 1)-dimensional Fokas equation. All these results show that the (4 + 1)-dimensional Fokas equation is a meaningful multidimensional extension of the KP and DS equations. The obtained results might be useful in diverse fields such as hydrodynamics, non-linear optics and photonics, ion-acoustic waves in plasmas, matter waves in Bose-Einstein condensates, and sound waves in ferromagnetic media.

nlin.SI↗