24,042 results on '"Nonlinear Sciences - Exactly Solvable and Integrable Systems"'
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2. On the Hilbert Space of the Chern-Simons Matrix Model, Deformed Double Current Algebra Action, and the Conformal Limit
- Author
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Hu, Sen, Li, Si, Ye, Dongheng, and Zhou, Yehao
- Subjects
Mathematical Physics ,High Energy Physics - Theory ,Mathematics - Quantum Algebra ,Mathematics - Representation Theory ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,81R12, 81R50, 16S38, 14A22 - Abstract
A Chern-Simons matrix model was proposed by Dorey, Tong, and Turner to describe non-Abelian fractional quantum Hall effect. In this paper we study the Hilbert space of the Chern-Simons matrix model from a geometric quantization point of view. We show that the Hilbert space of the Chern-Simons matrix model can be identified with the space of sections of a line bundle on the quiver variety associated to a framed Jordan quiver. We compute the character of the Hilbert space using localization technique. Using a natural isomorphism between vortex moduli space and a Beilinson-Drinfeld Schubert variety, we prove that the ground states wave functions are flat sections of a bundle of conformal blocks associated to a WZW model. In particular they solve a Knizhnik-Zamolodchikov equation. We show that there exists a natural action of the deformed double current algebra (DDCA) on the Hilbert space, moreover the action is irreducible. We define and study the conformal limit of the Chern-Simons matrix model. We show that the conformal limit of the Hilbert space is an irreducible integrable module of $\widehat{\mathfrak{gl}}(n)$ with level identified with the matrix model level. Moreover, we prove that $\widehat{\mathfrak{gl}}(n)$ generators can be obtained from scaling limits of matrix model operators, which settles a conjecture of Dorey-Tong-Turner. The key to the proof is the construction of a Yangian $Y(\mathfrak{gl}_n)$ action on the conformal limit of the Hilbert space, which we expect to be equivalent to the $Y(\mathfrak{gl}_n)$ action on the integrable $\widehat{\mathfrak{gl}}(n)$ modules constructed by Uglov. We also characterize eigenvectors and eigenvalues of the matrix model Hilbert space with respect to a maximal commutative subalgebra of Yangian., Comment: 80+9 pages. Comments are welcome
- Published
- 2024
3. Accelerating solutions of the Korteweg-de Vries equation
- Author
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Winkler, Maricarmen A. and Asenjo, Felipe A.
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
The Korteweg-de Vries equation is a fundamental nonlinear equation that describes solitons with constant velocity. On the contrary, here we show that this equation also presents accelerated wavepacket solutions. This behavior is achieved by putting the Korteweg-de Vries equation in terms of the Painlev\'e I equation. The accelerated waveform solutions are explored numerically showing their accelerated behavior explicitly., Comment: 4 pages, 2 figures
- Published
- 2024
4. Geometrically constrained sine-Gordon field: BPS solitons and their collisions
- Author
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da Hora, E., Pereira, L., Santos, C. dos, and Simas, F. C.
- Subjects
High Energy Physics - Theory ,Nonlinear Sciences - Pattern Formation and Solitons ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
We consider an enlarged $(1+1)$-dimensional model with two real scalar fields, $\phi$ and $\chi$ whose scalar potential $V(\phi,\chi)$ has a standard $\chi^4$ sector and a sine-Gordon one for $\phi$. These fields are coupled through a generalizing function $f(\chi)$ that appears in the scalar potential and controls the nontrivial dynamics of $\phi$. We minimize the effective energy via the implementation of the BPS technique. We then obtain the Bogomol'nyi bound for the energy and the first-order equations whose solutions saturate that bound. We solve these equations for a nontrivial $f(\chi)$. As the result, BPS kinks with internal structures emerge. They exhibit a two-kink profile. i.e. an effect due to geometrical constrictions. We consider the linear stability of these new configurations. In this sense, we study the existence of internal modes that play an important role during the scattering process. We then investigate the kink-antikink collisions, and present the numerical results for the most interesting cases. We also comment about their most relevant features., Comment: 30 pages, 12 figures. Suggestions are welcome
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- 2024
5. A class of exactly solvable Convection-Diffusion-Reaction equations in similarity form with intrinsic supersymmetry
- Author
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Ho, Choon-Lin
- Subjects
Mathematical Physics ,Condensed Matter - Statistical Mechanics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Quantum Physics - Abstract
In this work we would like to point out the possibility of generating a class of exactly solvable convection-diffusion-reaction equation in similarity form with intrinsic supersymmetry, i.e., the solution and the diffusion coefficient of the equation are supersymmetrically related through their similarity scaling forms., Comment: 7 pages, 2 figures
- Published
- 2024
6. Many-body density of states of bosonic and fermionic gases: a combinatorial approach
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Echter, Carolyn, Maier, Georg, Urbina, Juan-Diego, Lewenkopf, Caio, and Richter, Klaus
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Condensed Matter - Quantum Gases ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Quantum Physics - Abstract
We use a combinatorial approach to obtain exact expressions for the many-body density of states of fermionic and bosonic gases with equally spaced single-particle spectra. We identify a mapping that reveals a remarkable property, namely, fermionic and bosonic gases have the same many-body density of states, up to a shift corresponding to ground state energy. Additionally, we show that there is a regime, comprising the validity range of the Bethe approximation, where the many-body density of states becomes independent of the number of particles.
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- 2024
7. Delay ordinary differential equations: from Lagrangian approach to Hamiltonian approach
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Dorodnitsyn, Vladimir, Kozlov, Roman, and Meleshko, Sergey
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Mathematical Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
The paper suggests a Hamiltonian formulation for delay ordinary differential equations (DODEs). Such equations are related to DODEs with a Lagrangian formulation via a delay analog of the Legendre transformation. The Hamiltonian delay operator identity is established. It states the relationship for the invariance of a delay Hamiltonian functional, appropriate delay variational equations, and their conserved quantities. The identity is used to formulate a Noether-type theorem, which provides first integrals for Hamiltonian DODEs with symmetries. The relationship between the invariance of the delay Hamiltonian functional and the invariance of the delay variational equations is also examined. Several examples illustrate the theoretical results.
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- 2024
8. Symmetries of Toda type 3D lattices
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Habibullin, I. T. and Khakimova, A. R.
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Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
The duality between a class of the Davey-Stewartson type coupled systems and a class of two-dimensional Toda type lattices is discussed. For the recently found integrable lattice the hierarchy of symmetries is described. Second and third order symmetries are presented in explicit form. Corresponding coupled systems are given. An original method for constructing exact solutions to coupled systems is suggested based on the Darboux integrable reductions of the dressing chains. Some new solutions for coupled systems related to the Volterra lattice are presented as illustrative examples., Comment: International Conference <
>, Sochi--2024 - Published
- 2024
9. Spherical and hyperbolic orthogonal ring patterns: integrability and variational principles
- Author
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Bobenko, Alexander I.
- Subjects
Mathematics - Metric Geometry ,Mathematical Physics ,Mathematics - Complex Variables ,Mathematics - Differential Geometry ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
We introduce orthogonal ring patterns in the 2-sphere and in the hyperbolic plane, consisting of pairs of concentric circles, which generalize circle patterns. We show that their radii are described by a discrete integrable system. This is a special case of the master integrable equation Q4. The variational description is given in terms of elliptic generalizations of the dilogarithm function. They have the same convexity principles as their circle-pattern counterparts. This allows us to prove existence and uniqueness results for the Dirichlet and Neumann boundary value problems. Some examples are computed numerically. In the limit of small smoothly varying rings, one obtains harmonic maps to the sphere and to the hyperbolic plane. A close relation to discrete surfaces with constant mean curvature is explained., Comment: 29 pages, 11 figures
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- 2024
10. Bilinearization-reduction approach to the classical and nonlocal semi-discrete modified Korteweg-de Vries equations with nonzero backgrounds
- Author
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Deng, Xiao, Chen, Hongyang, Zhao, Song-Lin, and Ren, Guanlong
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
Quasi double Casoratian solutions are derived for a bilinear system reformulated from the coupled semi-discrete modified Korteweg-de Vries equations with nonzero backgrounds. These solutions, when applied with the classical and nonlocal reduction techniques, also satisfy the corresponding classical and nonlocal semi-discrete modified Korteweg-de Vries equations with nonzero backgrounds. They can be expressed explicitly, allowing for an easy investigation of the dynamics of systems. As illustrative examples, the dynamics of solitonic, periodic and rational solutions with a plane wave background are examined for the focusing semi-discrete Korteweg-de Vries equation and the defocusing reverse-space-time complex semi-discrete Korteweg-de Vries equation.
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- 2024
11. Correlation functions of the six-vertex IRF model and its quantum spin chain
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Tavares, T. S. and Ribeiro, G. A. P.
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Mathematical Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
We consider the interaction-round-a-face version of the isotropic six-vertex model. The associated spin chain is made of two coupled Heisenberg spin chains with different boundary twists. The phase diagram of the model and the long distance correlations were studied in [Nucl. Phys. B, 995 (2023) 116333]. Here, we compute the short-distance correlation functions of the model in the ground state for finite system sizes via non-linear integral equations and in the thermodynamic limit. This was possible since the model satisfies the face version of the discrete quantum Knizhnik-Zamolodchikov (qKZ) equation. A suitable ansatz for the density matrix is proposed in the form of a direct sum of two Heisenberg density matrices, which allows us to obtain the discrete functional equation for the two-site function $\omega(\lambda_1,\lambda_2)$. Thanks to the known results on the factorization of correlation functions of the Heisenberg chain, we are able to compute the density matrix of the IRF model for up to four sites and its associated spin chain for up to three sites., Comment: 25 pages, 4 figures, 1 table
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- 2024
12. Lax representations for the three-dimensional Euler--Helmholtz equation
- Author
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Morozov, Oleg I.
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
The paper is concerned with Lax representations for the three-dimensional Euler--Helmholtz equation. We show that the parameter in the Lax representation from Theorem 3 in [15] is non-removable. Then we present two new Lax representations with non-removable parameters.
- Published
- 2024
13. Auxiliary Field Deformations of (Semi-)Symmetric Space Sigma Models
- Author
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Bielli, Daniele, Ferko, Christian, Smith, Liam, and Tartaglino-Mazzucchelli, Gabriele
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High Energy Physics - Theory ,General Relativity and Quantum Cosmology ,Mathematical Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
We generalize the auxiliary field deformations of the principal chiral model (PCM) introduced in arXiv:2405.05899 and arXiv:2407.16338 to sigma models whose target manifolds are symmetric or semi-symmetric spaces, including a Wess-Zumino term in the latter case. This gives rise to a new infinite family of classically integrable $\mathbb{Z}_2$ and $\mathbb{Z}_4$ coset models of the form which are of interest in applications of integrability to worldsheet string theory and holography. We demonstrate that every theory in this infinite class admits a zero-curvature representation for its equations of motion by exhibiting a Lax connection., Comment: 52 pages; v2: reference updated
- Published
- 2024
14. Hietarinta's classification of $4\times 4$ constant Yang-Baxter operators using algebraic approach
- Author
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Maity, Somnath, Singh, Vivek Kumar, Padmanabhan, Pramod, and Korepin, Vladimir
- Subjects
High Energy Physics - Theory ,Condensed Matter - Statistical Mechanics ,Mathematical Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Quantum Physics - Abstract
Classifying Yang-Baxter operators is an essential first step in the study of the simulation of integrable quantum systems on quantum computers. One of the earliest initiatives was taken by Hietarinta in classifying constant Yang-Baxter solutions for a two-dimensional local Hilbert space (qubit representation). He obtained 11 families of invertible solutions, including the one generated by the permutation operator. While these methods work well for 4 by 4 solutions, they become cumbersome for higher dimensional representations. In this work, we overcome this restriction by constructing the constant Yang-Baxter solutions in a representation independent manner by using ans\"{a}tze from algebraic structures. We use four different algebraic structures that reproduce 10 of the 11 Hietarinta families when the qubit representation is chosen. The methods include a set of commuting operators, Clifford algebras, Temperley-Lieb algebras, and partition algebras. We do not obtain the $(2,2)$ Hietarinta class with these methods., Comment: 26 pages + References
- Published
- 2024
15. Interesting system of $3$ first-order recursions
- Author
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Calogero, Francesco
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematics - Dynamical Systems - Abstract
In this paper we firstly review how to \textit{explicitly} solve a system of $3$ \textit{first-order linear recursions }and outline the main properties of these solutions. Next, via a change of variables, we identify a class of systems of $3$ \textit{first-order nonlinear recursions} which also are \textit{explicitly solvable}. These systems might be of interest for practitioners in \textit{applied} sciences: they allow a complete display of their solutions, which may feature interesting behaviors, for instance be \textit{completely periodic} ("isochronous systems", if the independent variable $n=0,1,2,3...$is considered a \textit{ticking time}), or feature this property \textit{only asymptotically} (as\textit{\ }$n\rightarrow \infty $).
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- 2024
16. Integrability of polynomial vector fields and a dual problem
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Petek, Tatjana and Romanovski, Valery
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Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematics - Classical Analysis and ODEs ,Mathematics - Dynamical Systems - Abstract
We investigate the integrability of polynomial vector fields through the lens of duality in parameter spaces. We examine formal power series solutions anihilated by differential operators and explore the properties of the integrability variety in relation to the invariants of the associated Lie group. Our study extends to differential operators on affine algebraic varieties, highlighting the inartistic connection between these operators and local analytic first integrals. To illustrate the duality the case of quadratic vector fields is considered in detail.
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- 2024
17. Extended symmetry of higher Painlev\'e equations of even periodicity and their rational solutions
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Aratyn, Henrik, Gomes, José Francisco, Lobo, Gabriel Vieira, and Zimerman, Abraham Hirsz
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Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
The structure of extended affine Weyl symmetry group of higher Painlev\'e equations of $N$ periodicity depends on whether $N$ is even or odd. We find that for even $N$, the symmetry group ${\widehat A}^{(1)}_{N-1}$ contains the conventional B\"acklund transformations $s_j, j=1,{\ldots},N$, the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of $N$ points, which is a novel feature uncovered in this paper. The presence of reflection automorphisms is connected to existence of degenerated solutions and for $N=4$ we explicitly show how the reflection automorphisms around even points cause degeneracy of a class of rational solutions obtained on the orbit of translation operators of ${\widehat A}^{(1)}_{3}$. We obtain the closed expressions for solutions and their degenerated counterparts in terms of determinants of Kummer polynomials., Comment: 26 pages
- Published
- 2024
18. Hamiltonian models for the propagation of long gravity waves, higher-order KdV-type equations and integrability
- Author
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Ivanov, Rossen I.
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,76B15, 35Q35 - Abstract
A single incompressible, inviscid, irrotational fluid medium bounded above by a free surface is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet-Neumann operators. The equations for the surface waves are presented in Hamiltonian form. Specific scaling of the variables is selected which leads to a KdV approximation with higher order nonlinearities and dispersion (higher-order KdV-type equation, or HKdV). The HKdV is related to the known integrable PDEs with an explicit nonlinear and nonlocal transformation., Comment: 16 pages, 1 figure, book chapter
- Published
- 2024
- Full Text
- View/download PDF
19. On the long-wave approximation of solitary waves in cylindrical coordinates
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Hornick, James, Pelinovsky, Dmitry E., and Schneider, Guido
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Mathematics - Analysis of PDEs ,Mathematical Physics ,Mathematics - Classical Analysis and ODEs ,Nonlinear Sciences - Pattern Formation and Solitons ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
We address justification and solitary wave solutions of the cylindrical KdV equation which is formally derived as a long wave approximation of radially symmetric waves in a two-dimensional nonlinear dispersive system. For a regularized Boussinesq equation, we prove error estimates between true solutions of this equation and the associated cylindrical KdV approximation in the L2-based spaces. The justification result holds in the spatial dynamics formulation of the regularized Boussinesq equation. We also prove that the class of solitary wave solutions considered previously in the literature does not contain solutions in the L2-based spaces. This presents a serious obstacle in the applicability of the cylindrical KdV equation for modeling of radially symmetric solitary waves since the long wave approximation has to be performed separately in different space-time regions., Comment: 27 pages; 2 figures
- Published
- 2024
20. Stability of standing periodic waves in the massive Thirring model
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Cui, Shikun and Pelinovsky, Dmitry E.
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Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematical Physics ,Mathematics - Analysis of PDEs ,Mathematics - Dynamical Systems ,Nonlinear Sciences - Pattern Formation and Solitons - Abstract
We analyze the spectral stability of the standing periodic waves in the massive Thirring model in laboratory coordinates. Since solutions of the linearized MTM equation are related to the squared eigenfunctions of the linear Lax system, the spectral stability of the standing periodic waves can be studied by using their Lax spectrum. Standing periodic waves are classified based on eight eigenvalues which coincide with the endpoints of the spectral bands of the Lax spectrum. Combining analytical and numerical methods, we show that the standing periodic waves are spectrally stable if and only if the eight eigenvalues are located either on the imaginary axis or along the diagonals of the complex plane., Comment: 42 pages; 19 figures
- Published
- 2024
21. Generation Model of a Spatially Limited Vortex in a Stratified Unstable Atmosphere
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Onishchenko, O. G., Artekha, S. N., Feygin, F. Z., and Astafieva, N. M.
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Physics - Fluid Dynamics ,Mathematical Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Physics - Atmospheric and Oceanic Physics ,Physics - Geophysics - Abstract
This paper presents a new model for the generation of axisymmetric concentrated vortices. The solution of a nonlinear equation for internal gravity waves in an unstable stratified atmosphere is obtained and analyzed within the framework of ideal hydrodynamics. The corresponding expressions describing the dependences on the radius for the radial and vertical velocity components in the inner and outer regions of the vortex include combinations of Bessel functions and modified Bessel functions. The proposed new nonlinear analytical model makes it possible to study the structure and nonlinear dynamics of vortices in the radial and vertical regions. The vortex is limited in height. The maximum vertical velocity component is reached at a certain height. Below this height, radial flows converge towards the axis, and above it, an outflow occurs. The resulting instability in the stratified atmosphere leads to an increase in the radial and vertical velocity components according to the hyperbolic sine law, which turns into exponential growth. The characteristic growth time is determined by the inverse growth rate of the instability. The formation of vortices with finite velocity components, which increase with time, is analyzed. The radial structure of the azimuthal velocity is determined by the structure of the initial perturbation and can change with height. The maximum rotation is reached at a certain height. The growth of the azimuth velocity occurs according to a super-exponential law., Comment: 17 pages, 4 figures. arXiv admin note: text overlap with arXiv:2408.14210
- Published
- 2024
- Full Text
- View/download PDF
22. Magic Billiards: the Case of Elliptical Boundaries
- Author
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Dragović, Vladimir and Radnović, Milena
- Subjects
Mathematics - Dynamical Systems ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,37C83, 37J35, 37J39 - Abstract
In this work, we introduce a novel concept of magic billiard games and analyse their properties in the case of elliptical boundaries. We provide explicit conditions for periodicity in algebro-geometric, analytic, and polynomial forms. A topological description of those billiards is given using Fomenko graphs., Comment: 16 pages, 11 figures
- Published
- 2024
23. The integrable semi-discrete nonlinear Schr\'odinger equations with nonzero backgrounds: Bilinearization-reduction approach
- Author
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Deng, Xiao, Chen, Kui, Chen, Hongyang, and Zhang, Da-jun
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematical Physics - Abstract
In this paper the classical and nonlocal semi-discrete nonlinear Schr\"{o}dinger (sdNLS) equations with nonzero backgrounds are solved by means of the bilinearization-reduction approach. In the first step of this approach, the unreduced sdNLS system with a nonzero background is bilinearized and its solutions are presented in terms of quasi double Casoratians. Then, reduction techniques are implemented to deal with complex and nonlocal reductions, which yields solutions for the four classical and nonlocal sdNLS equations with a plane wave background or a hyperbolic function background. These solutions are expressed with explicit formulae and allow classifications according to canonical forms of certain spectral matrix. In particular, we present explicit formulae for general rogue waves for the classical focusing sdNLS equation. Some obtained solutions are analyzed and illustrated., Comment: 32 pages
- Published
- 2024
24. An integrable pseudospherical equation with pseudo-peakon solutions
- Author
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da Silva, Priscila Leal, Freire, Igor Leite, and Filho, Nazime Sales
- Subjects
Mathematical Physics ,Mathematics - Analysis of PDEs ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,35C08, 35D30, 35E05, 53C21 - Abstract
We study an integrable equation whose solutions define a triad of one-forms describing a surface with Gaussian curvature -1. We identify a local group of diffeomorphisms that preserve these solutions and establish conserved quantities. From the symmetries, we obtain invariant solutions that provide explicit metrics for the surfaces. These solutions are unbounded and often appear in mirrored pairs. We introduce the ``collage'' method, which uses conserved quantities to remove unbounded parts and smoothly join the solutions, leading to weak solutions consistent with the conserved quantities. As a result we get pseudo-peakons, which are smoother than Camassa-Holm peakons. Additionally, we apply a Miura-type transformation to relate our equation to the Degasperis-Procesi equation, allowing us to recover peakon and shock-peakon solutions for it from the solutions of the other equation., Comment: Parts of the original version concerning shock-peakons have been update, whereas references [9] and [10] have been added as well. We thank Professor Hans Lundmark for his valuable comments and suggestions made on the earlier version of the manuscript
- Published
- 2024
25. Newell-Whitehead-Segel Equation: An Exact, Generalized Solution
- Author
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Cundin, Luisiana
- Subjects
Mathematical Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
Derivation of an exact, general solution to Newell-Whitehead-Segel transient, nonlinear partial differential equation is provided for one to three dimensional cases, also, arbitrary power of nonlinearity.
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- 2024
26. Fr\'olicher-Nijenhuis geometry and integrable matrix PDE systems
- Author
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Müller-Hoissen, Folkert
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematical Physics ,Mathematics - Differential Geometry ,37K10, 35Qxx, 53Z05 - Abstract
Given two tensor fields of type (1,1) on a smooth n-dimensional manifold M, such that all their Fr\"olicher-Nijenhuis brackets vanish, the algebra of differential forms on M becomes a bi-differential graded algebra. As a consequence, there are partial differential equation (PDE) systems associated with it, which arise as the integrability condition of a system of linear equations and possess a binary Darboux transformation to generate exact solutions. We recover chiral models and potential forms of the self-dual Yang-Mills, as well as corresponding generalizations to higher than four dimensions, and obtain new integrable non-autonomous nonlinear matrix PDEs and corresponding systems., Comment: 20 pages
- Published
- 2024
27. Two-stage initial-value iterative physics-informed neural networks for simulating solitary waves of nonlinear wave equations
- Author
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Song, Jin, Zhong, Ming, Karniadakis, George Em, and Yan, Zhenya
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Physics - Computational Physics ,Computer Science - Artificial Intelligence ,Computer Science - Machine Learning ,Mathematical Physics ,Nonlinear Sciences - Pattern Formation and Solitons ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
We propose a new two-stage initial-value iterative neural network (IINN) algorithm for solitary wave computations of nonlinear wave equations based on traditional numerical iterative methods and physics-informed neural networks (PINNs). Specifically, the IINN framework consists of two subnetworks, one of which is used to fit a given initial value, and the other incorporates physical information and continues training on the basis of the first subnetwork. Importantly, the IINN method does not require any additional data information including boundary conditions, apart from the given initial value. Corresponding theoretical guarantees are provided to demonstrate the effectiveness of our IINN method. The proposed IINN method is efficiently applied to learn some types of solutions in different nonlinear wave equations, including the one-dimensional (1D) nonlinear Schr\"odinger equations (NLS) equation (with and without potentials), the 1D saturable NLS equation with PT -symmetric optical lattices, the 1D focusing-defocusing coupled NLS equations, the KdV equation, the two-dimensional (2D) NLS equation with potentials, the 2D amended GP equation with a potential, the (2+1)-dimensional KP equation, and the 3D NLS equation with a potential. These applications serve as evidence for the efficacy of our method. Finally, by comparing with the traditional methods, we demonstrate the advantages of the proposed IINN method., Comment: 25 pages, 17 figures
- Published
- 2024
28. Totally Nonnegative Pfaffian for Solitons in BKP Equation
- Author
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Chang, Jen Hsu
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
The BKP equation is obtained from the reduction of B type in the KP hierarchy under the orthogonal type transformation group for the KP equation. The skew Schur Q functions can be used to construct the Tau functions of solitons in the BKP equation. Then the totally nonnegative Pfaffian can be defined via the skew Schur Q functions to obtain nonsingular line solitons solution in the BKP equation. The totally nonnegative Pfaffians are investigated. The line solitons interact to form web like structure in the near field region and their resonances appearing in soliton graph could be investigated by the totally nonnegative Pfaffians., Comment: 22 pages, 3 firures
- Published
- 2024
29. Casting more light in the shadows: dual Somos-5 sequences
- Author
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Harrow, J. W. E. and Hone, A. N. W.
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Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematical Physics ,Mathematics - Number Theory - Abstract
Motivated by the search for an appropriate notion of a cluster superalgebra, incorporating Grassmann variables, Ovsienko and Tabachnikov considered the extension of various recurrence relations with the Laurent phenomenon to the ring of dual numbers. Furthermore, by iterating recurrences with specific numerical values, some particular well-known integer sequences, such as the Fibonacci sequence, Markoff numbers, and Somos sequences, were shown to produce associated ``shadow'' sequences when they were extended to the dual numbers. Here we consider the most general version of the Somos-5 recurrence defined over the ring of dual numbers $\mathbb{D}$ with complex coefficients, that is the ring $\mathbb{C}[\varepsilon]$ modulo the relation $\varepsilon^2=0$. We present three different ways to present the general solution of the initial value problem for Somos-5 and its shadow part: in analytic form, using the Weierstrass sigma function with arguments in $\mathbb{D}$; in terms of the solution of a linear difference equation; and using Hankel determinants constructed from $\mathbb{D}$-valued moments, via a connection with a Quispel-Roberts-Thompson (QRT) map over the dual numbers.
- Published
- 2024
30. Solitons in 4d Wess-Zumino-Witten models -- Towards unification of integrable systems --
- Author
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Hamanaka, Masashi and Huang, Shan-Chi
- Subjects
High Energy Physics - Theory ,Mathematical Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
We construct soliton solutions of the four-dimensional Wess-Zumino-Witten (4dWZW) model in the context of a unified theory of integrable systems with relation to the 4d/6d Chern-Simons theory. We calculate the action density of the solutions and find that the soliton solutions behave as the KP-type solitons, that is, the one-soliton solution has a localized action/energy density on a 3d hyperplane in 4-dimensions (soliton wall) and the n-soliton solution describes n intersecting soliton walls with phase shifts. We note that the Ward conjecture holds mostly in the split signature (+,+,-,-). Furthermore, the 4dWZW model describes the string field theory action of the open N=2 string theory in the four-dimensional space-time with the split signature and hence our soliton solutions would describe a new-type of physical objects in the N=2 string theory. We discuss instanton solutions in the 4dWZW model as well. Noncommutative extension and quantization of the unified theory of integrable systems are also discussed., Comment: 20 pages, 2 figures
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- 2024
31. Orthosymplectic $R$-matrices
- Author
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Hong, Kyungtak and Tsymbaliuk, Alexander
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Mathematics - Representation Theory ,High Energy Physics - Theory ,Mathematics - Quantum Algebra ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
We present a formula for trigonometric orthosymplectic $R$-matrices associated with any parity sequence. We further apply the Yang-Baxterization technique of [M.Ge, Y.Wu, K.Xue, "Explicit trigonometric Yang-Baxterization", Internat. J. Modern Phys. A 6 (1991), no.21, 3735-3779] to derive affine orthosymplectic $R$-matrices, generalizing [M.Mehta, K.Dancer, M.Gould, J.Links, "Generalized Perk-Schultz models: solutions of the Yang-Baxter equation associated with quantized orthosymplectic superalgebras", J. Phys. A 39 (2006), no.1, 17-26] that treated the case of the standard parity sequence., Comment: v1: 31pp, comments are welcome!
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- 2024
32. Asymptotic integrability and Hamilton theory of soliton's motion along large-scale background waves
- Author
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Kamchatnov, A. M.
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
We consider the problem of soliton-mean field interaction for the class of asymptotically integrable equations, where the notion of the complete integrability means that the Hamilton equations for the high-frequency wave packet propagation along a large-scale background wave have an integral of motion. Using the Stokes remark, we transform this integral to the integral for the soliton's equations of motion and then derive the Hamilton equations for the soliton's dynamics in a universal form expressed in terms of the Riemann invariants for the hydrodynamic background wave. The physical properties are specified by the concrete expressions for the Riemann invariants. The theory is illustrated by its application to the soliton's dynamics which is described by the Kaup-Boussinesq system., Comment: 8 pages
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- 2024
33. Volume Changing Symmetries by Matrix Product Operators
- Author
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Borsi, Márton and Pozsgay, Balázs
- Subjects
Condensed Matter - Statistical Mechanics ,High Energy Physics - Theory ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
We consider spin chain models with exotic symmetries that change the length of the spin chain. It is known that the XXZ Heisenberg spin chain at the supersymmetric point $\Delta=-1/2$ possesses such a symmetry: it is given by the supersymmetry generators, which change the length of the chain by one unit. We show that volume changing symmetries exist also in other spin chain models, and that they can be constructed using a special tensor network, which is a simple generalization of a Matrix Product Operator. As examples we consider the folded XXZ model and its perturbations, and also a new hopping model that is defined on constrained Hilbert spaces. We show that the volume changing symmetries are not related to integrability: the symmetries can survive even non-integrable perturbations. We also show that the known supersymmetry generator of the XXZ chain with $\Delta=-1/2$ can also be expressed as a generalized Matrix Product Operator., Comment: 17 pages, 7 figures
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- 2024
34. Asymmetry Amplification by a Nonadiabatic Passage through a Critical Point
- Author
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Tyagi, Bhavay, Suzuki, Fumika, Chernyak, Vladimir A., and Sinitsyn, Nikolai A.
- Subjects
Quantum Physics ,Mathematical Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
We propose and solve a minimal model of dynamic passage through a quantum second order phase transition in the presence of weak symmetry breaking interactions and no dissipation. The evolution eventually leads to a highly asymmetric state, no matter how weak the symmetry breaking term is. This suggests a potential mechanism for strong asymmetry in the production of particles with almost identical characteristics. The model's integrability also allows us to obtain exact Kibble-Zurek exponents for the scaling of the number of nonadiabatic excitations., Comment: 5 pages, 2 figures
- Published
- 2024
35. Toda lattice and Riemann type minimal surfaces
- Author
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Gui, Changfeng, Liu, Yong, Wang, Jun, and Yang, Wen
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematics - Analysis of PDEs ,Mathematics - Differential Geometry - Abstract
Toda lattice and minimal surfaces are related to each other through Allen-Cahn equation. In view of the structure of the solutions of the Toda lattice, we find new balancing configuration using techniques of integrable systems. This allows us to construct new singly periodic minimal surfaces. The genus of these minimal surfaces equals $j(j+1)/2-1$. They are natural generalization of the Riemann minimal surfaces, which have genus zero., Comment: 16 pages
- Published
- 2024
36. On a quadratic Poisson algebra and integrable Lotka-Volterra systems, with solutions in terms of Lambert's W function
- Author
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van der Kamp, Peter H., McLaren, D. I., and Quispel, G. R. W.
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematics - Dynamical Systems - Abstract
We study a class of integrable nonhomogeneous Lotka-Volterra systems whose quadratic terms defined by an antisymmetric matrix and whose linear terms consist of three blocks. We provide the Poisson algebra of their Darboux polynomials, and prove a contraction theorem. We then use these results to classify the systems according to the number of functionally independent (and, for some, commuting) integrals. We also establish separability/solvability by quadratures, given the solutions to the 2- and 3-dimensional systems, which we provide in terms of the Lambert W function., Comment: 19 pages, 1 table, 11 references
- Published
- 2024
37. On Symmetries of Finite Geometries
- Author
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Knill, Oliver
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Computer Science - Discrete Mathematics ,Mathematical Physics ,Mathematics - Dynamical Systems ,68RXX 37JXX - Abstract
The isospectral set of the Dirac matrix D=d+d* consists of orthogonal Q for which Q* D Q is an equivalent Dirac matrix. It can serve as the symmetry of a finite geometry G. The symmetry is a subset of the orthogonal group or unitary group and isospectral Lax deformations produce commuting flows d/dt D=[B(g(D)),D] on this symmetry space. In this note, we remark that like in the Toda case, D_t=Q_t* D_0 Q_t with exp(-t g(D))=Q_t R_t solves the Lax system., Comment: 10 pages, 2 figures, updated Mathematica code and additional graphics
- Published
- 2024
38. On an integrable discretization of the massive Thirring model in light-cone coordinates and the associated Yang-Baxter map
- Author
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Tsuchida, Takayuki
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematical Physics - Abstract
We propose a fully discrete analog of the massive Thirring model in light-cone coordinates by constructing its Lax-pair representation. This Lax-pair representation can also be used to define a new Yang-Baxter map, so we obtain a Yang-Baxter map that admits a continuous limit. We present most of the results for the general case where the dependent variables are matrix-valued., Comment: 28 pages; (v2) corrected inexact sentences on the continuous limit of a Yang-Baxter map
- Published
- 2024
39. Approximation of the thermodynamic limit of finite-gap solutions of the focusing NLS hierarchy by multisoliton solutions
- Author
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Jenkins, Robert and Tovbis, Alexander
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematical Physics ,35Q15, 35Q55 - Abstract
In this paper we approximate the thermodynamic limit of finite-gap solutions to any integrable equations in the focusing NLS hierarchy (fNLS, mKdV, ...) with an associated multisoliton solutions using the Riemann-Hilbert Problem approach. Moreover, we show that both the finite-gap and multisoliton solutions are approximated in the thermodynamic limit by a generalization of the primitive potentials introduced by V. Zakharov and his collaborators in the KdV context. Under certain assumptions on the spectral data for the finite gap potentials, we provide error estimates for the approximation on compact subsets of the $(x,t)$-plane.
- Published
- 2024
40. Soliton Dynamics of a Gauged Fokas-Lenells Equation Under Varying Effects of Dispersion and Nonlinearity
- Author
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Dutta, Riki, Talukdar, Sagardeep, Saharia, Gautam K., and Nandy, Sudipta
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Nonlinear Sciences - Pattern Formation and Solitons - Abstract
Davydova-Lashkin-Fokas-Lenells equation (DLFLE) is a gauged equivalent form of Fokas-Lenells equation (FLE) that addresses both spatio-temporal dispersion (STD) and nonlinear dispersion (ND) effects. The balance between those effects results a soliton which has always been an interesting topic in research due to its potential applicability as signal carrier in information technology. We have induced a variation to the dispersion effects and apply Hirota bilinear method to realise soliton solution of the proposed DLFLE and explore how the soliton dynamic behaves in accordance to the variation of the dispersion effects. The proposed equation is applicable for number of systems like ultrashort optical pulse, ioncyclotron plasma wave, Bose-Einstein condensate (BEC) matter-wave soliton under certain external fields, etc. The study on such systems under varying effects is very limited and we hope our work can benefit the researchers to understand soliton dynamics more and work on various other nonlinear fields under varying effects., Comment: Under review in journal
- Published
- 2024
41. The evolution of spectral data for nonlinear Klein-Gordon models
- Author
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Palheta, P. H. S., Assis, P. E. G., and Gonçalves, T. M. N.
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,High Energy Physics - Theory - Abstract
We investigate the effect of the breaking of integrability in the integrals of motion of a sine-Gordon-like system. The class of quasi-integrable models, discussed in the literature, inherits some of the integrable properties they are associated with. Our strategy, to investigate the problem through a deformation of the so-called inverse scattering method, has proven to be useful in the discussion of generic nonlinear Klein-Gordon potentials, as well as in particular cases presented here., Comment: 15 pages, 10 figures
- Published
- 2024
42. Integrability of Generalised Skew-Symmetric Replicator Equations via Graph Embeddings
- Author
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Visomirski, Matthew and Griffin, Christopher
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematical Physics ,Mathematics - Dynamical Systems - Abstract
It is known that there is a one-to-one mapping between oriented directed graphs and zero-sum replicator dynamics (Lotka-Volterra equations) and that furthermore these dynamics are Hamiltonian in an appropriately defined nonlinear Poisson bracket. In this paper, we investigate the problem of determining whether these dynamics are Liouville-Arnold integrable, building on prior work graph in graph decloning by Evripidou et al. [J. Phys. A., 55:325201, 2022] and graph embedding by Paik and Griffin [Phys. Rev. E. 107(5): L052202, 2024]. Using the embedding procedure from Paik and Griffin, we show (with certain caveats) that when a graph producing integrable dynamics is embedded in another graph producing integrable dynamics, the resulting graph structure also produces integrable dynamics. We also construct a new family of graph structures that produces integrable dynamics that does not arise either from embeddings or decloning. We use these results to classify the dynamics generated by almost all oriented directed graphs on six vertices, with three hold-out graphs that generate integrable dynamics and are not part of a natural taxonomy arising from known families and graph operations. These hold-out graphs suggest more structure is available to be found. Moreover, the work suggests that oriented directed graphs leading to integrable dynamics may be classifiable in an analogous way to the classification of finite simple groups, creating the possibility that there is a deep connection between integrable dynamics and combinatorial structures in graphs., Comment: 35 pages, 17 figures
- Published
- 2024
43. Auxiliary Field Sigma Models and Yang-Baxter Deformations
- Author
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Bielli, Daniele, Ferko, Christian, Smith, Liam, and Tartaglino-Mazzucchelli, Gabriele
- Subjects
High Energy Physics - Theory ,Mathematical Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
We combine the Yang-Baxter deformation with higher-spin auxiliary field deformations to construct a multi-parameter family of integrable deformations of the principal chiral model on a Lie group $G$ with semi-simple Lie algebra $\mathfrak{g}$. Our construction produces one integrable deformation for each pair $(\mathcal{R}, E)$, where $\mathcal{R}$ is an antisymmetric bilinear operator on $\mathfrak{g}$ obeying the modified classical Yang-Baxter equation and $E$ is a function of several variables. We show that every model in this family is (weakly) classically integrable by exhibiting a Lax representation for its equations of motion., Comment: 37 pages
- Published
- 2024
44. Toda Darboux transformations and vacuum expectation values
- Author
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Wang, Chengwei, Chen, Mengyao, and Cheng, Jipeng
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematical Physics ,35Q53, 37K10, 37K40 - Abstract
Determinant formulas for vacuum expectation values $\langle s+k+n-m,-s|e^{H(\mathbf{t})}\beta_m^{*}\cdots\beta_1^{*}\beta_n\cdots\beta_1g|k\rangle $ are given by using Toda Darboux transformations. Firstly notice that 2--Toda hierarchy can be viewed as the 2--component bosonizations of fermionic KP hierarchy, then two elementary Toda Darboux transformation operators $T_{+}(q)=\Lambda(q)\cdot\Delta\cdot q^{-1}$ and $T_{-}(r)=\Lambda^{-1}(r)^{-1}\cdot\Delta^{-1}\cdot r$ are constructed from the changes of Toda (adjoint) wave functions by using 2--component boson--fermion correspondence. Based on this, the above vacuum expectation values now can be realized as the successive applications of Toda Darboux transformations. So the corresponding determinant formulas can be derived from the determinant representations of Toda Darboux transformations. Finally by similar methods, we also give the determinant formulas for $\langle n-m|e^{\mathcal{H}(\mathbf{x})}\beta_m^{*}\cdots\beta_1^{*}\beta_n\cdots\beta_1g|k\rangle $ related with KP tau functions., Comment: 18 pages
- Published
- 2024
45. The modified Toda hierarchy
- Author
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Rui, Wenjuan, Guan, Wenchuang, Yang, Yi, and Cheng, Jipeng
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematical Physics ,35Q51, 37K10, 37K40 - Abstract
In this paper, modified Toda (mToda) equation is generalized to form an integrable hierarchy in the framework of Sato theory, which is therefore called mToda hierarchy. Inspired by the fact that Toda hierarchy is 2--component generalization of usual KP hierarchy, mToda hierarchy is constructed from bilinear equations of 2--component 1st modified KP hierarchy, where we provide the corresponding equivalence with Lax formulations. Then it is demonstrated that there are Miura links between Toda and mToda hierarchies, which means the definition of mToda hierarchy here is reasonable. Finally, Darboux transformations of the Toda and mToda hierarchies are also constructed by using the aforementioned Miura links., Comment: 23 pages
- Published
- 2024
46. Unusual Properties of Adiabatic Invariance in a Billiard Model Related to the Adiabatic Piston Problem
- Author
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Skinner, Joshua and Neishtadt, Anatoly
- Subjects
Physics - Classical Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,70H11 - Abstract
We consider the motion of two massive particles along a straight line. A lighter particle bounces back and forth between a heavier particle and a stationary wall, with all collisions being ideally elastic. It is known that if the lighter particle moves much faster than the heavier one, and the kinetic energies of the particles are of the same order, then the product of the speed of the lighter particle and the distance between the heavier particle and the wall is an adiabatic invariant: its value remains approximately constant over a long period. We show that the value of this adiabatic invariant, calculated at the collisions of the lighter particle with the wall, is a constant of motion (i.e., {an exact adiabatic invariant}). On the other hand, the value of this adiabatic invariant at the collisions between the particles slowly and monotonically decays with each collision. The model we consider is a highly simplified version of the classical adiabatic piston problem, where the lighter particle represents a gas particle, and the heavier particle represents the piston., Comment: 6 pages
- Published
- 2024
47. Triple critical point and emerging temperature scales in $SU(N)$ ferromagnetism at large $N$
- Author
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Polychronakos, Alexios P. and Sfetsos, Konstantinos
- Subjects
High Energy Physics - Theory ,Condensed Matter - Statistical Mechanics ,Mathematical Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems - Abstract
The non-Abelian ferromagnet recently introduced by the authors, consisting of atoms in the fundamental representation of $SU(N)$, is studied in the limit where $N$ becomes large and scales as the square root of the number of atoms $n$. This model exhibits additional phases, as well as two different temperature scales related by a factor $N\!/\!\ln N$. The paramagnetic phase splits into a "dense" and a "dilute" phase, separated by a third-order transition and leading to a triple critical point in the scale parameter $n/N^2$ and the temperature, while the ferromagnetic phase exhibits additional structure, and a new paramagnetic-ferromagnetic metastable phase appears at the larger temperature scale. These phases can coexist, becoming stable or metastable as temperature varies. A generalized model in which the number of $SU(N)$-equivalent states enters the partition function with a nontrivial weight, relevant, e.g., when there is gauge invariance in the system, is also studied and shown to manifest similar phases, with the dense-dilute phase transition becoming second-order in the fully gauge invariant case., Comment: 41 pages, 9 figures
- Published
- 2024
48. Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation
- Author
-
Hereman, Willy and Göktaş, Ünal
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematical Physics ,Physics - Fluid Dynamics ,Physics - Plasma Physics ,Primary 35Q51, 35Q53. Secondary 37K10, 37K40 - Abstract
Using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations thereby establishing their complete integrability. The Gardner equation is chosen as the key example for it comprises both the Korteweg-de Vries and modified Korteweg-de Vries (mKdV) equations. The Gardner and Miura transformations which connect these equations are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota's method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case) whereas the focusing Gardner equation has the standard elastically colliding solitons. The paper's aim is to provide a review of integrability properties and solutions of the Gardner equation and illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica but can be adapted for major computer algebra systems., Comment: Invited paper for a Special Issue on Symmetry Methods for Solving Differential Equations (Mehmet Pakdemirli, ed.) of Mathematical and Computational Applications (34 pages, 9 figures, 74 references)
- Published
- 2024
- Full Text
- View/download PDF
49. On the crystal limit of the q-difference sixth Painlev\'e equation
- Author
-
Joshi, Nalini and Roffelsen, Pieter
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematical Physics ,39A13, 33E17, 34M50, 39A45, 47B39, 14J26 - Abstract
We consider the Riemann-Hilbert correspondence associated with the $q$-difference sixth Painlev\'e equation in the crystal limit, i.e. $q\rightarrow 0$, and show two main results. First, the limit of this generically highly transcendental mapping is shown to exist. Second, we show that the limiting map is bi-rational and describe it explicitly., Comment: 21 pages, 1 figure
- Published
- 2024
50. The inverse scattering theory of Kadomtsev-Petviashvili II equations
- Author
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Wu, Derchyi
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematics - Analysis of PDEs ,35Q53, 35P25, 37K15 - Abstract
An overview of the inverse scattering theory of the Kadomtsev Petviashvili II equation with an emphasis on the inverse problem for perturbed KP multi line solitons is provided. It is shown that, despite additional algebraic or analytic techniques are introduced due to new singular structures, there exists a consistency of the inverse scattering theories for different backgrounds such as the vacuum, $1$-line solitons, and multi line solitons., Comment: arXiv admin note: text overlap with arXiv:2205.07432
- Published
- 2024
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