arXiv subjects

S. Y. Lou

Publications and source records attributed to S. Y. Lou.

At least 19 recordsLinked to original sources

arXiv2026

Painleve solitons of AKNS system and irrational algebraic solitons of NLS equations

A novel symmetry decomposition approach is introduced to derive the so-called ``Painlevé solitons'' of the Ablowitz-Kaup-Newell-Segur (AKNS) system. These Painlevé solitons propagate against a background governed by a Painlevé transcendent, establishing a fundamental generalization of the well-known elliptic solitons concept. We demonstrate that while elliptic solitons arise from the combination of translation invariance and square eigenfunction symmetry, a \textit{different} symmetry combination-scaling invariance, Galilean invariance, and square eigenfunction symmetry-generates ``Painlevé IV solitons'' for the AKNS system. This discovery represents a significant theoretical advance in integrable systems theory. By selecting special solutions of the Painlevé IV equation, we obtain explicit forms of several previously unknown classes of solutions for the AKNS system and the nonlinear Schrödinger (NLS) equation: irrational algebraic solitons, rational algebraic solitons, and parabolic cylindrical function solitons. These results dramatically expand the known solution landscape of one of the most important integrable models in mathematical physics, with broad implications for nonlinear wave phenomena across multiple physical disciplines including optics, Bose-Einstein condensates, and fluid dynamics.

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arXiv2026

Analyzing the relationship between infinite symmetries and $N$-soliton solutions in the AKNS system

This paper investigates the algebraic reduction of the infinite-dimensional symmetries of the Ablowitz-Kaup-Newell-Segur system when restricted to multi-soliton solution. By systematically analyzing, we demonstrate that the entire $K$-symmetry hierarchy collapses into a finite-dimensional module over the field of wave parameters, spanned by elementary center-translation generators. Higher order $K$-symmetries are explicitly reconstructed as linear combinations of these basis vectors. In contrast, $τ$-symmetries resist such decomposition on pure soliton backgrounds, however, they become finite-dimensional once the underlying solution space is extended to the full multi-wave manifold, which carries more independent wave parameters. We construct an explicit basis consisting of four fundamental symmetry vector fields, two lowest $K$-symmetries and two primary $τ$-symmetries, in terms of which all higher $τ$-symmetries are uniquely expressible as linear combinations of these symmetry vector fields. These findings not only clarify the algebraic structure of infinite symmetries on special solution, but also provide an algorithmic framework for deriving exact multi-wave solutions of integrable systems.

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arXiv2025

Residual Symmetry Reductions and Painlevé Solitons

This letter introduces the novel concept of Painlevé solitons -- waves arising from the interaction between Painlevé waves and solitons in integrable systems. Painlevé solitons may also be viewed as solitons propagating against a Painlevé wave background, in analogy with the established notion of elliptic solitons, which refer to solitons on an elliptic wave background. By employing a novel symmetry decomposition method aided by nonlocal residual symmetries, we explicitly construct (extended) Painlevé II solitons for the Korteweg-de Vries (KdV) equation and (extended) Painlevé IV solitons for the Boussinesq equation.

nlin.SI↗

arXiv2025

Permutation--invariant Niven numbers

This paper introduces permutation-invariant Niven numbers--a novel class of Niven numbers where all digit permutations (with leading zeros automatically ignored) must retain the Niven property. We demonstrate that there exist infinitely many such numbers and that their magnitude is unbounded. Furthermore, we present an exhaustive search method for identifying permutation--invariant Niven numbers.

math.CO↗

arXiv2025

Peakons and pseudo-peakons of higher order b-family equations

This paper explores the rich structure of peakon and pseudo-peakon solutions for a class of higher-order $b$-family equations, referred to as the $J$-th $b$-family ($J$-bF) equations. We propose several conjectures concerning the weak solutions of these equations, including a $b$-independent pseudo-peakon solution, a $b$-independent peakon solution, and a $b$-dependent peakon solution. These conjectures are analytically verified for $J \leq 14$ and/or $J \leq 9$ using the computer algebra software MAPLE. The $b$-independent pseudo-peakon solution is a 3rd-order pseudo-peakon for general arbitrary constants, with higher-order pseudo-peakons derived under specific parameter constraints. Additionally, we identify both $b$-independent and $b$-dependent peakon solutions, highlighting their distinct properties and the nuanced relationship between the parameters $b$ and $J$. The existence of these solutions underscores the rich dynamical structure of the $J$-bF equations and generalizes previous results for lower-order equations. Future research directions include higher-order generalizations, rigorous proofs of the conjectures, interactions between different types of peakons and pseudo-peakons, stability analysis, and potential physical applications. These advancements significantly contribute to the understanding of peakon systems and their broader implications in mathematics and physics.

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arXiv2025

Exploring the depths of symmetry in the mKdV equation: physical interpretations and multi-wave solutions

This manuscript embarks on an in-depth exploration of the modified Korteweg-de Vries (mKdV) equation, with a particular emphasis on unraveling the intricate structure of its infinite symmetries and their physical interpretations. Central to this investigation are the $K$-symmetries and $τ$-symmetries, which are delineated by a recursive relationship and constitute an infinite ensemble that underpins the conservation laws. We engage with an existing symmetry conjecture, which posits that the currently identified symmetries represent a subset of a more expansive, yet to be unearthed, set. This conjecture is substantiated through an analysis of the soliton solutions associated with the mKdV equation, demonstrating that these symmetries can be decomposed into linear combinations of center and wave number translation symmetries. Further, by imposing an infinite sequence of symmetry constraints, it becomes feasible to derive exact multi-wave solutions. This methodology, predicated on the proposed symmetry conjecture, facilitates the extraction of exact solutions, encompassing complexiton, breather, multi-soliton solutions, among others.

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arXiv2024

Progresses on some open problems related to infinitely many symmetries

The quest to reveal the physical essence of the infinitely many symmetries and conservation laws that are intrinsic to integrable systems has historically posed a significant challenge at the confluence of physics and mathematics. This scholarly investigation delves into five open problems related to these boundless symmetries within integrable systems by scrutinizing their multi-wave solutions, employing a fresh analytical methodology. For a specified integrable system, there exist various categories of $n$-wave solutions. Each sub-wave comprising the $n$-wave solution may possess free parameters, including center, width, and periodic parameters. It is evident that these solutions are translation invariant with respect to all these free parameters. We postulate that the entirety of the recognized infinite symmetries merely constitute linear combinations of these finite wave parameter translation symmetries. The conjecture intimates that the currently known infinitely many symmetries are not exhaustive, and an indeterminate number of symmetries remain to be discovered. This conjecture further indicates that by imposing an infinite array of symmetry constraints, it becomes feasible to derive exact multi-wave solutions. By considering the renowned KdV equation and the Burgers equation as simple examples, the conjecture is substantiated for the $n$-soliton solutions. It is unequivocal that any linear combination of the wave parameter translation symmetries retains its status as a symmetry associated with the particular solution. This observation suggests that by introducing a ren-variable and a ren-symmetric derivative which serve as generalizations of the Grassmann variable and the super derivative, it may be feasible to unify classical integrable systems, supersymmetric integrable systems, and ren-symmetric integrable systems within a cohesive hierarchical framework.

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arXiv2024

From one to infinity: symmetries of integrable systems

Integrable systems constitute an essential part of modern physics. Traditionally, to approve a model is integrable one has to find its infinitely many symmetries or conserved quantities. In this letter, taking the well known Korteweg-de Vries and Boussinesq equations as examples, we show that it is enough to find only one nonlocal key-symmetry to guarantee the integrability. Starting from the nonlocal key-symmetry, recursion operator(s) and then infinitely many symmetries and Lax pairs can be successfully found.

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arXiv2023

Extensions of dark KdV equations: nonhomogeneous classifications, bosonizations of fermionic systems and supersymmetric dark systems

Dark equations are defined as some kinds of integrable couplings with some fields being homogeneously and linearly coupled to others. In this paper, dark equations are extended in several aspects. Taking the Korteweg-de Vrise (KdV) equation as an example, the dark KdV systems are extended to nonhomogenous forms, nonlinear couplings and graded linear cases. The two-component nonhomogeneous linear coupled dark KdV systems are completely classified. The nonlinear coupled dark KdV systems may be obtained through the decompositions from higher dimensional integrable systems like the B-type KP equation. Graded linear coupled dark KdV systems may be produced by introducing dark parameters (including the Grassmann parameters) to usual integrable systems. Especially, applying the bosonization approach to the integrable systems with fermion fields such as the supersymmetric integrable systems and super-integrable models, infinitely many graded linear dark systems can be generated. Finally, the dark KdV systems are extended to supersymmetric ones. The full classifications for the supersymmetric dark KdV systems are obtained related to two types of usual supersymmetric KdV equations.

nlin.SI↗

arXiv2023

Ren-integrable and ren-symmetric integrable systems

A new type of symmetry, ren-symmetry describing anyon physics and the corresponding topological physics, is proposed. Ren-symmetry is a generalization of super-symmetry which is widely applied in super-symmetric physics such as the super-symmetric quantum mechanics, super-symmetric gravity, super-symmetric string theory, super-symmetric integrable systems and so on. The super-symmetry and Grassmann-number are, in some sense, the dual conceptions, which turns out that these conceptions coincide for the ren situation, that is, a similar conception of ren-number is devised to ren-symmetry. In particular, some basic results of the ren-number and ren-symmetry are exposed which allow one to derive, in principle, some new types of integrable systems including ren-integrable models and ren-symmetric integrable systems. Training examples of ren-integrable KdV type systems and ren-symmetric KdV equations are explicitly given.

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arXiv2023

Higher dimensional integrable deformations of the modified KdV equation

The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability. The well-known modified KdV equation is a prototypical example of integrable evolution equations in one spatial dimension. Do there exist integrable analogs of modified KdV equation in higher spatial dimensions? In what follows, we present a positive answer to this question. In particular, rewriting the (1+1)-dimensional integrable modified KdV equation in conservation forms and adding deformation mappings during the process allow one to construct higher dimensional integrable equations. Further, we illustrate this idea with examples from the modified KdV hierarchy, also present the Lax pairs of these higher dimensional integrable evolution equations.

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arXiv2023

Deformation Conjecture: Deforming Lower Dimensional Integrable Systems to Higher Dimensional Ones by Using Conservation Laws

Utilizing some conservation laws of (1+1)-dimensional integrable local evolution systems, it is conjectured that higher dimensional integrable equations may be regularly constructed by a deformation algorithm. The algorithm can be applied to Lax pairs and higher order flows. In other words, if the original lower dimensional model is Lax integrable (possesses Lax pairs) and symmetry integrable (possesses infinitely many higher order symmetries), then the deformed higher order systems are also Lax integrable and symmetry integrable. For concreteness, the deformation algorithm is applied to the usual (1+1)-dimensional KdV equation and the (1+1)-dimensional AKNS system (including nonlinear NLS equation as a special example). It is interesting that the deformed (3+1)-dimensional KdV equation is also an extension of the (1+1)-dimensional Harry-Dym (HD) type equations which are reciprocal links of the (1+1)-dimensional KdV equation. The Lax pairs of the (3+1)-dimensional KdV-HD system and the (2+1)-dimensional AKNS system are explicitly given. The higher order symmetries, i.e., the whole (3+1)-dimensional KdV-HD hierarchy, are also explicitly obtained via the deformation algorithm. The single soliton solution of the (3+1)-dimensional KdV-HD equation is implicitly given. Because of the effects of the deformation, the symmetric soliton shape of the usual KdV equation is no longer conserved and deformed to be asymmetric and/or multi-valued. The deformation conjecture is correct for almost all the known (1+1)-dimensional integrable local evolution systems and we have not yet found any counter-example so far. The introduction of a large number of (D+1)-dimensional integrable systems of this paper explores a serious challenge to all mathematicians and theoretical physicists because the traditional methods are no longer directly valid to solve these integrable equations.

nlin.SI↗

arXiv2022

Decomposition solutions and Bäcklund transformations of the BKP and CKP equations

In this paper, we define the modified formal variable separation approach and show how it determines, in a remarkably simple manner, the decomposition solutions, the Bäcklund transformations, the Lax pair, and the linear superposition solution of the B-type Kadomtsev-Petviashvili equation. Also, the decomposition solutions, the Bäcklund transformation and the Lax pair relating to the C-type Kadomtsev-Petviashvili equation is obtain by the same technique. This indicates that the decomposition may provide a description of integrable behavior in nonlinear systems, while, at the same time, establishing an efficient method for determining relationships between the particular systems.

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arXiv2022

Multi-component decompositions, linear superpositions, and new nonlinear integrable coupled KdV-type systems

The existence of decompositions of the nonlinear integrable systems not only permits us to establish so-called linear superposition solutions but also to derive new nonlinear integrable coupled systems. Restricting our attention to the single component decompositions of the potential BKP hierarchy, we obtain that suitable linear superpositions of some decomposition solutions still satisfy the same equations. In parallel, successful attempts are made by multi-component decompositions of the potential BKP hierarchy to construct linear superposition solutions and new nonlinear integrable coupled KdV-type systems that a change of dependent variables cannot decouple.

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arXiv2022

Linear superposition in the general heavenly equation

Evidently, the linear superposition principle can not be exactly established as a general principle in the presence of nonlinearity, and, at the first glance, there is no expectation for it to hold even approximately. In this letter, it is shown that the balance of different nonlinear effects describes what linear superpositions may occur in nonlinear systems. The heavenly equations are of significance in several scientific fields, especially in relativity, gravity, field theory, and fluid dynamics. A special type of implicit shock wave solution with three two-dimensional arbitrary functions of the general heavenly equation is revealed. Restrict the two-dimensional arbitrary functions to some types of one-dimensional arbitrary functions, it is found that the nonlinear effects can be balanced such that the "impossible" linear superposition solutions can be nontrivially constituted to new solutions of the general heavenly equation.

math-ph↗

arXiv2022

Integrable nonlinear Klein-Gordon systems with $\mathcal{PT}$ nonlocality and/or space-time exchange nonlocality

In additional to the parity ($\mathcal{P}$) symmetric, time reversal ($\mathcal{T}$) symmetric, and $\mathcal{PT}$ symmetric nonlocal integrable systems, some other types of nonlocal integrable Klein-Gordon models with the space-time exchange nonlocality and the moving nolocality are proposed. The Lax pairs of the established nonlinear nonlocal Klein-Gordon equations are explicitly given. A special soliton solution, composed of $\mathcal{PT}$-symmetric part and $\mathcal{PT}$-antisymmetric part, is illustrated with the shape change.

nlin.SI↗

arXiv2021

Special decompositions and linear superpositions of nonlinear systems: BKP and dispersionless BKP equations

The existence of decomposition solutions of the well-known nonlinear BKP hierarchy is explored. It is shown that these decompositions provide simple and interesting relationships between classical integrable systems and the BKP hierarchy. Further, some special decomposition solutions display a rare property: they can be linearly superposed. With the emphasis on the case of the fifth BKP equation, the structure characteristic having linear superposition solutions is analyzed. Finally, we obtain similar superposed solutions in the dispersionless BKP hierarchy.

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arXiv2021

Duality of positive and negative integrable hierarchies via relativistically invariant fields

It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual relations, some continuous and discrete integrable positive hierarchies such as the potential modified Korteweg-de Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL hierarchy and the second heavenly hierarchies are converted to the integrable negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy, the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the dualities between the recursion operators and their inverses. In (2+1)-dimensional cases, the positive/negative hierarchy dualities are explicitly shown by using the formal series symmetry approach, the mastersymmetry method and the relativistic invariance of the duality relations. For the 4-dimensional heavenly system, the duality problem is studied firstly by formal series symmetry approach. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach so that the duality problem can also be studied by means of the recursion operators.

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