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Journal articles on the topic 'Isospectral transformations'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 February 2022

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1

Duarte, Pedro, and Maria Joana Torres. "Eigenvectors of isospectral graph transformations." Linear Algebra and its Applications 474 (June 2015): 110–23. http://dx.doi.org/10.1016/j.laa.2015.01.038.

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2

Bunimovich, L. A., and B. Z. Webb. "Isospectral compression and other useful isospectral transformations of dynamical networks." Chaos: An Interdisciplinary Journal of Nonlinear Science 22, no. 3 (September 2012): 033118. http://dx.doi.org/10.1063/1.4739253.

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3

Chou, Tian, and Zhang Youjin. "Backlund transformations for the isospectral and non-isospectral MKdV hierarchies." Journal of Physics A: Mathematical and General 23, no. 13 (July 7, 1990): 2867–77. http://dx.doi.org/10.1088/0305-4470/23/13/024.

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4

Krajewska, Agnieszka, Alexander Ushveridze, and Zbigniew Walczak. "Anti-Isospectral Transformations in Quantum Mechanics." Modern Physics Letters A 12, no. 17 (June 7, 1997): 1225–34. http://dx.doi.org/10.1142/s0217732397001242.

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In this letter we demonstrate that there exists a remarkable class of quantum models admitting Z2 anti-isospectral transformations which change the form of the potential and invert the sign of a certain finite set of energy levels. These transformations help one to construct interesting classes of non-monic polynomials which are mutually orthogonal on two different intervals with two different weight functions.

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5

Guliyev, Namig J. "Essentially isospectral transformations and their applications." Annali di Matematica Pura ed Applicata (1923 -) 199, no. 4 (November 27, 2019): 1621–48. http://dx.doi.org/10.1007/s10231-019-00934-w.

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6

WANG, QINMOU, UDAY P. SUKHATME, WAI-YEE KEUNG, and TOM D. IMBO. "SOLITONS FROM SUPERSYMMETRY." Modern Physics Letters A 05, no. 07 (March 20, 1990): 525–30. http://dx.doi.org/10.1142/s0217732390000603.

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Supersymmetry transformations constitute a new, powerful technique of generating families of isospectral potentials. We show that isospectral families of reflectionless potentials provide surprisingly simple expressions for the pure multi-soliton solutions of the KdV and other nonlinear evolution equations.

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7

Gottlieb, H. P. W. "Transformations between isospectral membranes yield conformal maps." IMA Journal of Applied Mathematics 70, no. 6 (December 1, 2005): 748–52. http://dx.doi.org/10.1093/imamat/hxh067.

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8

Rosenhouse, Avia, and Jacob Katriel. "Isospectral transformations in relativistic and nonrelativistic mechanics." Letters in Mathematical Physics 13, no. 2 (February 1987): 141–46. http://dx.doi.org/10.1007/bf00955203.

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9

FENG, QIAN, YI-TIAN GAO, XIANG-HUA MENG, XIN YU, ZHI-YUAN SUN, TAO XU, and BO TIAN. "N-SOLITON-LIKE SOLUTIONS AND BÄCKLUND TRANSFORMATIONS FOR A NON-ISOSPECTRAL AND VARIABLE-COEFFICIENT MODIFIED KORTEWEG-DE VRIES EQUATION." International Journal of Modern Physics B 25, no. 05 (February 20, 2011): 723–33. http://dx.doi.org/10.1142/s0217979211058043.

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A non-isospectral and variable-coefficient modified Korteweg–de Vries (mKdV) equation is investigated in this paper. Starting from the Ablowitz–Kaup–Newell–Segur procedure, the Lax pair is established and the Bäcklund transformation in original variables is also derived. By a dependent variable transformation, the non-isospectral and variable-coefficient mKdV equation is transformed into bilinear equations, by virtue of which the N-soliton-like solution is obtained. In addition, the bilinear Bäcklund transformation gives a one-soliton-like solution from a vacuum one. Furthermore, the N-soliton-like solution in the Wronskian form is constructed and verified via the Wronskian technique.

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10

BINDING, PAUL A., PATRICK J. BROWNE, and BRUCE A. WATSON. "TRANSFORMATIONS BETWEEN STURM–LIOUVILLE PROBLEMS WITH EIGENVALUE DEPENDENT AND INDEPENDENT BOUNDARY CONDITIONS." Bulletin of the London Mathematical Society 33, no. 6 (November 2001): 749–57. http://dx.doi.org/10.1112/s0024609301008177.

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Explicit relationships are given connecting ‘almost’ isospectral Sturm–Liouville problems with eigen-value dependent, and independent, boundary conditions, respectively. Application is made to various direct and inverse spectral questions.

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11

Viviani, Milo. "A minimal-variable symplectic method for isospectral flows." BIT Numerical Mathematics 60, no. 3 (December 16, 2019): 741–58. http://dx.doi.org/10.1007/s10543-019-00792-1.

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AbstractIsospectral flows are abundant in mathematical physics; the rigid body, the the Toda lattice, the Brockett flow, the Heisenberg spin chain, and point vortex dynamics, to mention but a few. Their connection on the one hand with integrable systems and, on the other, with Lie–Poisson systems motivates the research for optimal numerical schemes to solve them. Several works about numerical methods to integrate isospectral flows have produced a large varieties of solutions to this problem. However, many of these algorithms are not intrinsically defined in the space where the equations take place and/or rely on computationally heavy transformations. In the literature, only few examples of numerical methods avoiding these issues are known, for instance, the spherical midpoint method on $${{\mathfrak {s}}}{{\mathfrak {o}}}(3)$$ s o ( 3 ) . In this paper we introduce a new minimal-variable, second order, numerical integrator for isospectral flows intrinsically defined on quadratic Lie algebras and symmetric matrices. The algorithm is isospectral for general isospectral flows and Lie–Poisson preserving when the isospectral flow is Hamiltonian. The simplicity of the scheme, together with its structure-preserving properties, makes it a competitive alternative to those already present in literature.

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12

SATTINGER, D. H. "FLAT CONNECTIONS AND INTEGRABLE SYSTEMS." International Journal of Modern Physics B 03, no. 02 (February 1989): 211–33. http://dx.doi.org/10.1142/s0217979289000166.

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The theory of completely integrable systems based on n × n first order isospectral problems is reviewed. The topics discussed are the inverse scattering theory for n × n systems; hierarchies of completely integrable Hamiltonian systems based on semi-simple Lie algebras; flat connections and gauge transformations; and the gauge theory of Bäcklund transformations.

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13

Deng, Shu-fang. "Darboux transformations for the isospectral and nonisospectral mKP equation." Physica A: Statistical Mechanics and its Applications 382, no. 2 (August 2007): 487–93. http://dx.doi.org/10.1016/j.physa.2007.04.009.

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14

Adler, M., and P. Vanmoerbeke. "Birkhoff Strata, Bäcklund Transformations, and Regularization of Isospectral Operators." Advances in Mathematics 108, no. 1 (September 1994): 140–204. http://dx.doi.org/10.1006/aima.1994.1070.

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15

Tian, C., and Y. J. Zhang. "Bäcklund transformations for the isospectral and nonisospectral AKNS hierarchies." Journal of Mathematical Physics 31, no. 9 (September 1990): 2150–54. http://dx.doi.org/10.1063/1.528617.

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16

Zhang, Ning, and Tiecheng Xia. "A Hierarchy of Lattice Soliton Equations Associated with a New Discrete Eigenvalue Problem and Darboux Transformations." International Journal of Nonlinear Sciences and Numerical Simulation 16, no. 7-8 (December 1, 2015): 301–6. http://dx.doi.org/10.1515/ijnsns-2014-0119.

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AbstractBy considering a new discrete isospectral eigenvalue problem, a hierarchy of integrable positive and negative lattice models is derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. And the equation in the resulting hierarchy is integrable in Liouville sense. Further, a Darboux transformation is established for the typical equations by using gauge transformations of Lax pairs, from which the exact solutions are given.

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17

Khare, Avinash, and Bhabani Prasad Mandal. "Anti-isospectral transformations, orthogonal polynomials, and quasi-exactly solvable problems." Journal of Mathematical Physics 39, no. 6 (June 1998): 3476–86. http://dx.doi.org/10.1063/1.532442.

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18

Zhang, Sheng, and Wang Di. "A Toda lattice hierarchy with variable coefficients and its multi-wave solutions." Thermal Science 18, no. 5 (2014): 1563–66. http://dx.doi.org/10.2298/tsci1405563z.

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Starting from the Toda spectral problem, a new Toda lattice hierarchy of isospectral equations with variable coefficients is constructed through the discrete zero curvature equation. In order to solve one special case of the derived Toda lattice hierarchy, a series of appropriate transformations are utilized. As a result, a new uniform formula of N-wave solutions is obtained.

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19

Bunimovich, L. A., and B. Z. Webb. "Isospectral graph transformations, spectral equivalence, and global stability of dynamical networks." Nonlinearity 25, no. 1 (December 15, 2011): 211–54. http://dx.doi.org/10.1088/0951-7715/25/1/211.

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20

Kong, O. C. W., and P. C. W. Fung. "Isospectral Hamiltonians from inverse scattering: I. Transformations in a united framework." Inverse Problems 5, no. 5 (October 1, 1989): 783–98. http://dx.doi.org/10.1088/0266-5611/5/5/008.

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21

Kong, O. C. W., and P. C. W. Fung. "Isospectral Hamiltonians from inverse scattering: II. Transformations and singular-soliton potentials." Inverse Problems 5, no. 5 (October 1, 1989): 799–815. http://dx.doi.org/10.1088/0266-5611/5/5/009.

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22

Katriel, Jacob, and Avia Rosenhouse. "Classical limit of the Korteweg-de Vries hierarchy of isospectral transformations." Physical Review D 32, no. 4 (August 15, 1985): 884–90. http://dx.doi.org/10.1103/physrevd.32.884.

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23

Bunimovich, Leonid, and Longmei Shu. "Generalized eigenvectors of isospectral transformations, spectral equivalence and reconstruction of original networks." Linear Algebra and its Applications 551 (August 2018): 104–24. http://dx.doi.org/10.1016/j.laa.2018.04.007.

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24

Levi, D., and O. Ragnisco. "Non-isospectral deformations and Darboux transformations for the third-order spectral problem." Inverse Problems 4, no. 3 (August 1, 1988): 815–28. http://dx.doi.org/10.1088/0266-5611/4/3/016.

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25

Pakuliak, Stanislav, and Sergei Sergeev. "Quantum relativistic Toda chain at root of unity: isospectrality, modifiedQ-operator, and functional Bethe ansatz." International Journal of Mathematics and Mathematical Sciences 31, no. 9 (2002): 513–53. http://dx.doi.org/10.1155/s0161171202105059.

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We investigate anN-state spin model called quantum relativistic Toda chain and based on the unitary finite-dimensional representations of the Weyl algebra withqbeingNth primitive root of unity. Parameters of the finite-dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter'sQ-operators. The classical counterpart of the modifiedQ-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modifiedQ-operators.

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26

Estévez, P. G., and C. Sardón. "Miura-reciprocal transformations for non-isospectral Camassa-Holm hierarchies in 2 + 1 dimensions." Journal of Nonlinear Mathematical Physics 20, no. 4 (October 2, 2013): 552–64. http://dx.doi.org/10.1080/14029251.2013.868268.

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27

Rosenhouse, Avia, and Jacob Katriel. "The Korteweg–de Vries hierarchy of isospectral transformations: Towards a general explicit expression." Journal of Mathematical Physics 28, no. 6 (June 1987): 1344–50. http://dx.doi.org/10.1063/1.527536.

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28

Loomba, Shally. "The management of matter rogue waves in F = 1 spinor Bose–Einstein condensates." International Journal of Modern Physics B 29, no. 19 (July 21, 2015): 1550125. http://dx.doi.org/10.1142/s0217979215501258.

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We have reported the matter rogue wave solutions for the nonautonomous three coupled Gross–Pitaevskii (GP) equation which governs the pulse propagation in spin-1 Bose–Einstein condensates (BECs) with time modulated nonlinearities. The system under consideration has attractive mean-field interactions and ferromagnetic spin-exchange interactions. The exact solutions have been worked out by using similarity and scaling transformations. We have depicted the controllable center characteristics of rogue waves for the kink-type spin-exchange interactions. Additionally, we have also discussed the management of rogue waves for the hyperbolic form of spin-exchange interactions by invoking isospectral Hamiltonian deformation technique.

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29

ZHANG, YUFENG, HONWAH TAM, and WEI JIANG. "TWO ENLARGED LOOP ALGEBRAS FOR OBTAINING NEW INTEGRABLE HIERARCHIES." International Journal of Modern Physics B 25, no. 19 (July 30, 2011): 2637–56. http://dx.doi.org/10.1142/s0217979211100904.

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Taking a loop algebra [Formula: see text] we obtain an integrable soliton hierarchy which is similar to the well-known Kaup–Newell (KN) hierarchy, but it is not. We call it a modified KN (mKN) hierarchy. Then two new enlarged loop algebras of the loop algebra [Formula: see text] are established, respectively, which are used to establish isospectral problems. Thus, two various types of integrable soliton-equation hierarchies along with multi-component potential functions are obtained. Their Hamiltonian structures are also obtained by the variational identity. The second hierarchy is integrable couplings of the mKN hierarchy. This paper provides a clue for generating loop algebras, specially, gives an approach for producing new integrable systems. If we obtain a new soliton hierarchy, we could deduce its symmetries, conserved laws, Darboux transformations, soliton solutions and so on. Hence, the way presented in the paper is an important aspect to obtain new integrable systems in soliton theory.

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30

CASAHORRÁN, J. "A NEW SUPERSYMMETRIC VERSION OF THE ABRAHAM-MOSES METHOD FOR SYMMETRIC POTENTIALS." Reviews in Mathematical Physics 08, no. 05 (July 1996): 655–68. http://dx.doi.org/10.1142/s0129055x96000226.

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Starting from the one-dimensional Schrodinger equation with symmetric potential Vs(x), a general method is presented in order to obtain a family of partially isospectral hamiltonians. Arguments concerning supersymmetric transformations, factorization procedures and Riccati equations are invoked along the article. As a result of the appearance of singular superpotentials, the physical meaning of our method can be summarized as follows: only the odd wave-functions of the original potential Vs(x) are transported via supersymmetry into the Hilbert space associated with the partner Vp(x). In such a case the degeneracy of energy levels is partially broken. Supersymmetry is neither exact nor spontaneously broken but realizes itself acting on wave functions vanishing at x=0. While the domain of the original hamiltonian H extends along the whole real axis, the susy partner Hp reduces to the half-line (x≤0 or x≥0). To illustrate how the procedure works in practice we resort to a symmetric potential in the Posch-Teller class containing both discrete and continuous spectra.

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31

SUN, YE-PENG, and GUO-FU YU. "BÄCKLUND TRANSFORMATION OF A NON-ISOSPECTRAL KPESCS AND ITS NONLINEAR COUPLED SYSTEM." Modern Physics Letters B 23, no. 30 (December 10, 2009): 3581–95. http://dx.doi.org/10.1142/s0217984909021636.

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A non-isospectral Kadomtsev–Petviashvili equation with self-consistent sources (KPESCS) is first derived by using the source generation procedure. Moreover, its bilinear Bäcklund transformation is obtained. Furthermore, a new nonlinear coupled system of the non-isospectral KPESCS and its exact solutions are established. The new coupled system cannot only be reduced to the non-isospectral KPESCS but can also be reduced to the coupled non-isospectral KP equation.

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32

Zhang, Xiangzhi, Yufeng Zhang, and Jianqin Mei. "Some symmetries and similarity solutions of the long-water wave hierarchy." Modern Physics Letters B 33, no. 34 (December 10, 2019): 1950430. http://dx.doi.org/10.1142/s021798491950430x.

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We introduce isospectral and non-isospectral Lax pairs, then apply the Tu scheme to generate the isospectral integrable hierarchy and the non-isospectral hierarchy, whose Hamiltonian structure, hereditary operator, symmetries are followed to obtain. In addition, a kind of Bäcklund transformation of the long-water wave hierarchy for the isospectral hierarchy is constructed. Through reductions of the isospectral hierarchy, we again get the long-water wave system whose similarity solutions, nonlinear self-adjointness and the non-invariant solutions are investigated, respectively, by the use of symmetry analysis. Finally, we make use of the weight method of the variables appearing in the long-water wave system to analyze the conservation laws of the system.

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33

Bhat K, Srivatsa, and Ranjan Ganguli. "Isospectrals of non-uniform Rayleigh beams with respect to their uniform counterparts." Royal Society Open Science 5, no. 2 (February 2018): 171717. http://dx.doi.org/10.1098/rsos.171717.

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In this paper, we look for non-uniform Rayleigh beams isospectral to a given uniform Rayleigh beam. Isospectral systems are those that have the same spectral properties, i.e. the same free vibration natural frequencies for a given boundary condition. A transformation is proposed that converts the fourth-order governing differential equation of non-uniform Rayleigh beam into a uniform Rayleigh beam. If the coefficients of the transformed equation match with those of the uniform beam equation, then the non-uniform beam is isospectral to the given uniform beam. The boundary-condition configuration should be preserved under this transformation. We present the constraints under which the boundary configurations will remain unchanged. Frequency equivalence of the non-uniform beams and the uniform beam is confirmed by the finite-element method. For the considered cases, examples of beams having a rectangular cross section are presented to show the application of our analysis.

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34

Gao, Xudong, and Sheng Zhang. "Inverse scattering transform for a new non-isospectral integrable non-linear AKNS model." Thermal Science 21, suppl. 1 (2017): 153–60. http://dx.doi.org/10.2298/tsci17s1153g.

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Constructing integrable systems and solving non-linear partial differential equations are important and interesting in non-linear science. In this paper, Ablowitz-Kaup-Newell-Segur (AKNS)?s linear isospectral problem and its accompanied time evolution equation are first generalized by embedding a new non-isospectral parameter whose varying with time obeys an arbitrary smooth enough function of the spectral parameter. Based on the generalized AKNS linear problem and its evolution equation, a new non-isospectral Lax integrable non-linear AKNS model is then derived. Furthermore, exact solutions of the derived AKNS model is obtained by extending the inverse scattering transformation method with new time-varying spectral parameter. In the case of reflectinless potentials, explicit n-soliton solutions are finally formulated through the obtained exact solutions.

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35

DING, HAI-YONG, XI-XIANG XU, HONG-XIANG YANG, and XIANG TIAN. "A NEW HIERARCHY OF DISCRETE INTEGRABLE MODEL AND ITS DARBOUX TRANSFORMATION." Modern Physics Letters B 21, no. 04 (February 10, 2007): 189–97. http://dx.doi.org/10.1142/s0217984907012360.

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A new isospectral problem is proposed and the corresponding integrable equation hierarchy is given. A Darboux transformation (DT) is derived to obtain the exact solutions for the typical lattice soliton equations.

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36

Yong, Xuelin, Yajing Fan, Yehui Huang, Wen-Xiu Ma, and Jing Tian. "Darboux transformation and solitons for an integrable nonautonomous nonlinear integro-differential Schrödinger equation." Modern Physics Letters B 31, no. 30 (October 26, 2017): 1750276. http://dx.doi.org/10.1142/s0217984917502761.

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By modifying the scheme for an isospectral problem, the non-isospectral Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy is constructed via allowing the time varying spectrum. In this paper, we consider an integrable nonautonomous nonlinear integro-differential Schrödinger equation discussed before in “Multi-soliton management by the integrable nonautonomous nonlinear integro-differential Schrödinger equation” [Y. J. Zhang, D. Zhao and H. G. Luo, Ann. Phys. 350 (2014) 112]. We first analyze the integrability conditions and identify the model. Second, we modify the existing Darboux transformation (DT) for such a non-isospectral problem. Third, the nonautonomous soliton solutions are obtained via the resulting DT and basic properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients. In the process, a technique by selecting appropriate spectral parameters instead of the variable inhomogeneities is employed to realize a different type of one-soliton management. Several novel optical solitons are constructed and their features are shown by some specific figures. In addition, four kinds of the special localized two-soliton solutions are obtained. The solitonic excitations localized both in space and time, which exhibit the feature of the so-called rogue waves but with a zero background, are discussed.

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37

BAGCHI, B., and A. GANGULY. "GENERATING ISOSPECTRAL HAMILTONIANS FROM A MODIFIED CRUM–DARBOUX TRANSFORMATION." International Journal of Modern Physics A 13, no. 21 (August 20, 1998): 3711–21. http://dx.doi.org/10.1142/s0217751x98001748.

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Using supersymmetry as a tool we set up a procedure to generate a sequence of pairwise isospectral potentials from a modified Crum–Darboux transformation. We show that such a transformation gives rise to shifted superpotentials much similar to what we encounter in a modified factorization scheme. We also explore some solvable systems to illustrate the applicability of our scheme.

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BAGCHI, B., A. GANGULY, D. BHAUMIK, and A. MITRA. "HIGHER DERIVATIVE SUPERSYMMETRY, A MODIFIED CRUM–DARBOUX TRANSFORMATION AND COHERENT STATE." Modern Physics Letters A 14, no. 01 (January 10, 1999): 27–34. http://dx.doi.org/10.1142/s0217732399000055.

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Inspired by the possibility of factorizing the second derivative interwining operators using a modified form of the well-known Crum–Darboux transformation, we present a scheme for generating a new pair of isospectral potentials. We also consider the interesting problem of constructing coherent states for such factorizable operators.

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39

Dong, Ruan, Wang Xue-Jin, Huang Wei-Cheng, and Sun Hong-Zhou. "Semiunitary Transformation and Isospectral Hamiltonians in Arbitrary Dimensional Space." Communications in Theoretical Physics 36, no. 1 (July 15, 2001): 25–28. http://dx.doi.org/10.1088/0253-6102/36/1/25.

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40

Chen, Yu-Qi, Bo Tian, Qi-Xing Qu, He Li, Xue-Hui Zhao, He-Yuan Tian, and Meng Wang. "Ablowitz–Kaup–Newell–Segur system, conservation laws and Bäcklund transformation of a variable-coefficient Korteweg–de Vries equation in plasma physics, fluid dynamics or atmospheric science." International Journal of Modern Physics B 34, no. 25 (September 8, 2020): 2050226. http://dx.doi.org/10.1142/s0217979220502264.

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For a variable-coefficient Korteweg–de Vries equation in a lake/sea, two-layer liquid, atmospheric flow, cylindrical plasma or interactionless plasma, in this paper, we derive the bilinear Bäcklund transformation, non-isospectral Ablowitz–Kaup–Newell–Segur system and infinite conservation laws for the wave amplitude under certain constraints among the external force, dissipation, nonlinearity, dispersion and perturbation.

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41

SINHA, A., and P. ROY. "DARBOUX TRANSFORMATION FOR THE ONE-DIMENSIONAL STATIONARY DIRAC EQUATION WITH NON-HERMITIAN INTERACTION." International Journal of Modern Physics A 21, no. 28n29 (November 20, 2006): 5807–22. http://dx.doi.org/10.1142/s0217751x0603312x.

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The Darboux algorithm is applied to an exactly solvable one-dimensional stationary Dirac equation, with non-Hermitian, pseudoscalar interaction V0(x). This generates a hierarchy of exactly solvable Dirac Hamiltonians, [Formula: see text], defined by new non-Hermitian interactions V1(x), which are also pseudoscalar. It is shown that [Formula: see text] are isospectral to the initial Hamiltonian h0, except for certain missing states.

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42

ZHANG, YUFENG, and HONWAH TAM. "APPLICATIONS OF THE LIE ALGEBRA gl(2)." Modern Physics Letters B 23, no. 14 (June 10, 2009): 1763–70. http://dx.doi.org/10.1142/s0217984909019922.

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With the help of a subalgebra of the Lie algebra gl (2) (still denoted by gl (2)), a non-isospectral evolution-equation hierarchy is obtained, whose reduction is similar to that given by Li. Employing the Lie algebra gl (2) again, we work out nonlinear evolution equations with exponential terms and produce their Bäcklund Transformation as well as some exact solutions.

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43

ZHANG, YUFENG, HONWAH TAM, and JING ZHAO. "GENERALIZED mKdV EQUATION, LIOUVILLE EQUATION, SINE-GORDON EQUATION AND SINH-GORDON EQUATION AS WELL AS A FORMAL BÄCKLUND TRANSFORMATION." International Journal of Modern Physics B 25, no. 18 (July 20, 2011): 2449–60. http://dx.doi.org/10.1142/s0217979211101387.

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A Lie algebra which consists of linear combinations of one basis of the Lie algebra A1 is presented for which an isospectral Lax pair is exhibited. By using the zero curvature equation, the generalized mKdV equation, Liouville equation and sine-Gordon equation, sinh-Gordon equation are generated via polynomial expansions. Finally, we investigate a kind of formal Bäcklund transformation for the generalized sine-Gordon equation. The explicit Bäcklund transformation of the standard sine-Gordon equation is presented. The other equations given in the paper are obtained similarly.

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44

Li, Hong-Zhe, Bo Tian, Rui Guo, Yu-Shan Xue, and Feng-Hua Qi. "Gauge transformation between the first-order nonisospectral and isospectral Heisenberg hierarchies." Applied Mathematics and Computation 218, no. 15 (April 2012): 7694–99. http://dx.doi.org/10.1016/j.amc.2012.01.018.

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45

Li, Li. "An isospectral transformation and the related integrable hierarchies of soliton equations." Chaos, Solitons & Fractals 38, no. 2 (October 2008): 598–607. http://dx.doi.org/10.1016/j.chaos.2007.01.011.

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46

ZHU, LI-LI, JUN DU, XIAO-YAN MA, and SHENG-JU SANG. "INTEGRABLE COUPLING SYSTEMS OF HAMILTONIAN LATTICE EQUATIONS BY SEMI-DIRECT SUMS OF LIE ALGEBRAS." Modern Physics Letters B 24, no. 07 (March 20, 2010): 681–94. http://dx.doi.org/10.1142/s0217984910022755.

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Abstract:

By considering a discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations are derived. The relation to the Toda type lattice is achieved by variable transformation. With the help of Tu scheme, the Hamiltonian structure of the resulting lattice hierarchy is constructed. The Liouville integrability is then demonstrated. Semi-direct sum of Lie algebras is proposed to construct discrete integrable couplings. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

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47

Yurova, A. A., A. V. Yurov, and M. Rudnev. "Darboux transformation for classical acoustic spectral problem." International Journal of Mathematics and Mathematical Sciences 2003, no. 49 (2003): 3123–42. http://dx.doi.org/10.1155/s016117120330105x.

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We study discrete isospectral symmetries for the classical acoustic spectral problem in spatial dimensions one and two by developing a Darboux (Moutard) transformation formalism for this problem. The procedure follows steps similar to those for the Schrödinger operator. However, there is no one-to-one correspondence between the two problems. The technique developed enables one to construct new families of integrable potentials for the acoustic problem, in addition to those already known. The acoustic problem produces a nonlinear Harry Dym PDE. Using the technique, we reproduce a pair of simple soliton solutions of this equation. These solutions are further used to construct a new positon solution for this PDE. Furthermore, using the dressing-chain approach, we build a modified Harry Dym equation together with its LA pair. As an application, we construct some singular and nonsingular integrable potentials (dielectric permitivity) for the Maxwell equations in a 2D inhomogeneous medium.

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48

Yadav, Rajesh Kumar, Nisha Kumari, Avinash Khare, and Bhabani Prasad Mandal. "RATIONALLY EXTENDED SHAPE INVARIANT POTENTIALS IN ARBITRARY D DIMENSIONS ASSOCIATED WITH EXCEPTIONAL Xm POLYNOMIALS." Acta Polytechnica 57, no. 6 (December 30, 2017): 477. http://dx.doi.org/10.14311/ap.2017.57.0477.

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Rationally extended shape invariant potentials in arbitrary D-dimensions are obtained by using point canonical transformation (PCT) method. The bound-state solutions of these exactly solvable potentials can be written in terms of <em>X<sub>m</sub></em> Laguerre or <em>X<sub>m</sub></em> Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property.

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XU, XIAO-GE, XIANG-HUA MENG, FU-WEI SUN, and YI-TIAN GAO. "INTEGRABLE PROPERTIES FOR A GENERALIZED NON-ISOSPECTRAL AND VARIABLE-COEFFICIENT KORTEWEG–DE VRIES MODEL." Modern Physics Letters B 24, no. 10 (April 20, 2010): 1023–32. http://dx.doi.org/10.1142/s0217984910022949.

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Applicable in fluid dynamics and plasmas, a generalized variable-coefficient Korteweg–de Vries (vcKdV) equation is investigated analytically employing the Hirota bilinear method in this paper. The bilinear form for such a model is derived through a dependent variable transformation. Based on the bilinear form, the integrable properties such as the N-solitonic solution, the Bäcklund transformation and the Lax pair for the vcKdV equation are obtained. Additionally, it is shown that the bilinear Bäcklund transformation can turn into the one denoted in the original variables.

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50

Zhou, Lingjun. "Darboux transformation for the non-isospectral AKNS hierarchy and its asymptotic property." Physics Letters A 372, no. 34 (August 2008): 5523–28. http://dx.doi.org/10.1016/j.physleta.2008.06.072.

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