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.
科特韦格-德弗里斯方程是一个描述匀速孤子的基本非线性方程。相反，我们在这里证明，该方程还呈现出加速波包解。这种行为是通过将 Korteweg-de Vries 方程置于 Painlev'e I 方程中来实现的。我们对加速波形解进行了数值探索，明确显示了它们的加速行为。
{"title":"Accelerating solutions of the Korteweg-de Vries equation","authors":"Maricarmen A. Winkler, Felipe A. Asenjo","doi":"arxiv-2409.10426","DOIUrl":"https://doi.org/arxiv-2409.10426","url":null,"abstract":"The Korteweg-de Vries equation is a fundamental nonlinear equation that\u0000describes solitons with constant velocity. On the contrary, here we show that\u0000this equation also presents accelerated wavepacket solutions. This behavior is\u0000achieved by putting the Korteweg-de Vries equation in terms of the Painlev'e I\u0000equation. The accelerated waveform solutions are explored numerically showing\u0000their accelerated behavior explicitly.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Symmetries of Toda type 3D lattices","authors":"I. T. Habibullin, A. R. Khakimova","doi":"arxiv-2409.07017","DOIUrl":"https://doi.org/arxiv-2409.07017","url":null,"abstract":"The duality between a class of the Davey-Stewartson type coupled systems and\u0000a class of two-dimensional Toda type lattices is discussed. For the recently\u0000found integrable lattice the hierarchy of symmetries is described. Second and\u0000third order symmetries are presented in explicit form. Corresponding coupled\u0000systems are given. An original method for constructing exact solutions to\u0000coupled systems is suggested based on the Darboux integrable reductions of the\u0000dressing chains. Some new solutions for coupled systems related to the Volterra\u0000lattice are presented as illustrative examples.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Bilinearization-reduction approach to the classical and nonlocal semi-discrete modified Korteweg-de Vries equations with nonzero backgrounds","authors":"Xiao Deng, Hongyang Chen, Song-Lin Zhao, Guanlong Ren","doi":"arxiv-2409.06168","DOIUrl":"https://doi.org/arxiv-2409.06168","url":null,"abstract":"Quasi double Casoratian solutions are derived for a bilinear system\u0000reformulated from the coupled semi-discrete modified Korteweg-de Vries\u0000equations with nonzero backgrounds. These solutions, when applied with the\u0000classical and nonlocal reduction techniques, also satisfy the corresponding\u0000classical and nonlocal semi-discrete modified Korteweg-de Vries equations with\u0000nonzero backgrounds. They can be expressed explicitly, allowing for an easy\u0000investigation of the dynamics of systems. As illustrative examples, the\u0000dynamics of solitonic, periodic and rational solutions with a plane wave\u0000background are examined for the focusing semi-discrete Korteweg-de Vries\u0000equation and the defocusing reverse-space-time complex semi-discrete\u0000Korteweg-de Vries equation.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Lax representations for the three-dimensional Euler--Helmholtz equation","authors":"Oleg I. Morozov","doi":"arxiv-2409.05752","DOIUrl":"https://doi.org/arxiv-2409.05752","url":null,"abstract":"The paper is concerned with Lax representations for the three-dimensional\u0000Euler--Helmholtz equation. We show that the parameter in the Lax representation\u0000from Theorem 3 in [15] is non-removable. Then we present two new Lax\u0000representations with non-removable parameters.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Henrik Aratyn, José Francisco Gomes, Gabriel Vieira Lobo, Abraham Hirsz Zimerman
The structure of the extended affine Weyl symmetry group of higher Painlev'e equations with periodicity $N$ varies depending on whether $N$ is even or odd. For even $N$, the symmetry group ${widehat A}^{(1)}_{N-1}$ includes not only the conventional B"acklund transformations $s_j, j=1,{ldots},N$, and the group of automorphisms consisting of cycling permutations but also incorporates %encompasses an additional expansion of the group of automorphisms by embedding %featuring in this group the reflections on a periodic circle of $N$ points. This latter aspect is a novel feature revealed in this paper. The presence of reflection automorphisms is linked to the existence of degenerated solutions. Specifically, for $N=4$ we explicitly demonstrate how the reflection automorphisms around even points induce degeneracy in a class of rational solutions obtained on the orbit of translation operators of ${widehat A}^{(1)}_{3}$. We provide closed-form expressions for both the solutions and their degenerated counterparts, given in terms of determinants of Kummer polynomials.
{"title":"Extended symmetry of higher Painlevé equations of even periodicity and their rational solutions","authors":"Henrik Aratyn, José Francisco Gomes, Gabriel Vieira Lobo, Abraham Hirsz Zimerman","doi":"arxiv-2409.03534","DOIUrl":"https://doi.org/arxiv-2409.03534","url":null,"abstract":"The structure of the extended affine Weyl symmetry group of higher Painlev'e\u0000equations with periodicity $N$ varies depending on whether $N$ is even or odd.\u0000For even $N$, the symmetry group ${widehat A}^{(1)}_{N-1}$ includes not only\u0000the conventional B\"acklund transformations $s_j, j=1,{ldots},N$, and the\u0000group of automorphisms consisting of cycling permutations but also incorporates\u0000%encompasses an additional expansion of the group of automorphisms by embedding\u0000%featuring in this group the reflections on a periodic circle of $N$ points.\u0000This latter aspect is a novel feature revealed in this paper. The presence of reflection automorphisms is linked to the existence of\u0000degenerated solutions. Specifically, for $N=4$ we explicitly demonstrate how\u0000the reflection automorphisms around even points induce degeneracy in a class of\u0000rational solutions obtained on the orbit of translation operators of ${widehat\u0000A}^{(1)}_{3}$. We provide closed-form expressions for both the solutions and\u0000their degenerated counterparts, given in terms of determinants of Kummer\u0000polynomials.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Hamiltonian models for the propagation of long gravity waves, higher-order KdV-type equations and integrability","authors":"Rossen I. Ivanov","doi":"arxiv-2409.03091","DOIUrl":"https://doi.org/arxiv-2409.03091","url":null,"abstract":"A single incompressible, inviscid, irrotational fluid medium bounded above by\u0000a free surface is considered. The Hamiltonian of the system is expressed in\u0000terms of the so-called Dirichlet-Neumann operators. The equations for the\u0000surface waves are presented in Hamiltonian form. Specific scaling of the\u0000variables is selected which leads to a KdV approximation with higher order\u0000nonlinearities and dispersion (higher-order KdV-type equation, or HKdV). The\u0000HKdV is related to the known integrable PDEs with an explicit nonlinear and\u0000nonlocal transformation.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Totally Nonnegative Pfaffian for Solitons in BKP Equation","authors":"Jen Hsu Chang","doi":"arxiv-2409.00711","DOIUrl":"https://doi.org/arxiv-2409.00711","url":null,"abstract":"The BKP equation is obtained from the reduction of B type in the KP hierarchy\u0000under the orthogonal type transformation group for the KP equation. The skew\u0000Schur Q functions can be used to construct the Tau functions of solitons in the\u0000BKP equation. Then the totally nonnegative Pfaffian can be defined via the skew\u0000Schur Q functions to obtain nonsingular line solitons solution in the BKP\u0000equation. The totally nonnegative Pfaffians are investigated. The line solitons\u0000interact to form web like structure in the near field region and their\u0000resonances appearing in soliton graph could be investigated by the totally\u0000nonnegative Pfaffians.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Volume Changing Symmetries by Matrix Product Operators","authors":"Márton Borsi, Balázs Pozsgay","doi":"arxiv-2408.15659","DOIUrl":"https://doi.org/arxiv-2408.15659","url":null,"abstract":"We consider spin chain models with exotic symmetries that change the length\u0000of the spin chain. It is known that the XXZ Heisenberg spin chain at the\u0000supersymmetric point $Delta=-1/2$ possesses such a symmetry: it is given by\u0000the supersymmetry generators, which change the length of the chain by one unit.\u0000We show that volume changing symmetries exist also in other spin chain models,\u0000and that they can be constructed using a special tensor network, which is a\u0000simple generalization of a Matrix Product Operator. As examples we consider the\u0000folded XXZ model and its perturbations, and also a new hopping model that is\u0000defined on constrained Hilbert spaces. We show that the volume changing\u0000symmetries are not related to integrability: the symmetries can survive even\u0000non-integrable perturbations. We also show that the known supersymmetry\u0000generator of the XXZ chain with $Delta=-1/2$ can also be expressed as a\u0000generalized Matrix Product Operator.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Asymptotic integrability and Hamilton theory of soliton's motion along large-scale background waves","authors":"A. M. Kamchatnov","doi":"arxiv-2408.15662","DOIUrl":"https://doi.org/arxiv-2408.15662","url":null,"abstract":"We consider the problem of soliton-mean field interaction for the class of\u0000asymptotically integrable equations, where the notion of the complete\u0000integrability means that the Hamilton equations for the high-frequency wave\u0000packet propagation along a large-scale background wave have an integral of\u0000motion. Using the Stokes remark, we transform this integral to the integral for\u0000the soliton's equations of motion and then derive the Hamilton equations for\u0000the soliton's dynamics in a universal form expressed in terms of the Riemann\u0000invariants for the hydrodynamic background wave. The physical properties are\u0000specified by the concrete expressions for the Riemann invariants. The theory is\u0000illustrated by its application to the soliton's dynamics which is described by\u0000the Kaup-Boussinesq system.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. H. S. Palheta, P. E. G. Assis, T. M. N. Gonçalves
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.
{"title":"The evolution of spectral data for nonlinear Klein-Gordon models","authors":"P. H. S. Palheta, P. E. G. Assis, T. M. N. Gonçalves","doi":"arxiv-2408.10101","DOIUrl":"https://doi.org/arxiv-2408.10101","url":null,"abstract":"We investigate the effect of the breaking of integrability in the integrals\u0000of motion of a sine-Gordon-like system. The class of quasi-integrable models,\u0000discussed in the literature, inherits some of the integrable properties they\u0000are associated with. Our strategy, to investigate the problem through a\u0000deformation of the so-called inverse scattering method, has proven to be useful\u0000in the discussion of generic nonlinear Klein-Gordon potentials, as well as in\u0000particular cases presented here.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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