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  • 1
    Electronic Resource
    Electronic Resource
    A Hierarchy of Multidimensional Hénon-Heiles Systems (2000)
    Springer
    Acta mathematica sinica 16 (2000), S. 527-534 
    Details
    ISSN: 1000-9574
    Keywords: Key words Higher-order constrained flow, Multidimensional Hénon-Heiles systems, Lax representation, Jacobi inversion problem ; 1991MR Subject Classification 58F07, 35Q53
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A hierarchy of multidimensional Hénon-Heiles (M-H-H) systems are constructed via the x- and t n -higher-order-constrained flows of KdV hierarchy. The Lax representation for the M-H-H hierarchy is determined from the adjoint representation of the auxiliary linear problem for the KdV hierarchy. By using the Lax representation the classical Poisson structure and r-matrix for the hierarchy are found and the Jacobi inversion problem for the hierarchy is constructed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/PL00011559
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  • 2
    Electronic Resource
    Electronic Resource
    Separation of variables for soliton equations via their binary constrained flows (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 6526-6557 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Binary constrained flows of soliton equations admitting 2×2 Lax matrices have 2N degrees of freedom, which is twice as many degrees of freedom than in the case of monoconstrained flows. By using the normal method, their Lax matrices directly give rise to first N pairs of canonical separated variables for their separation of variables. We propose a new method to introduce the other N pairs of canonical separated variables and additional separated equations. The Jacobi inversion problems for binary constrained flows are established. Finally, the factorization of soliton equations by two commuting binary constrained flows and the separability of binary constrained flows enable us to construct the Jacobi inversion problems for some soliton hierarchies. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533105
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  • 3
    Electronic Resource
    Electronic Resource
    Integrable Hamiltonian systems related to the polynomial eigenvalue problem (1990)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 2835-2839 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The independent integrals of motion in involution for the Hamiltonian system related to the second-order polynomial eigenvalue problem are constructed by using relevant recursion formula. The hierarchy of Hamiltonian systems obtained from the above problem and the time part of the Lax pair are shown to be completely integrable and they are shown to commute with each other. Furthermore, their solution solves the evolution equation associated with the Lax pair.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528987
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  • 4
    Electronic Resource
    Electronic Resource
    Integration of the soliton hierarchy with self-consistent sources (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 5453-5489 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In contrast with the soliton equations, the evolution of the eigenfunctions in the Lax representation of soliton equation with self-consistent sources (SESCS) possesses singularity. We present a general method to treat the singularity to determine the evolution of scattering data. The AKNS hierarchy with self-consistent sources, the MKdV hierarchy with self-consistent sources, the nonlinear Schrödinger equation hierarchy with self-consistent sources, the Kaup–Newell hierarchy with self-consistent sources and the derivative nonlinear Schrödinger equation hierarchy with self-consistent sources are integrated directly by using the inverse scattering method. The N soliton solutions for some SESCS are presented. It is shown that the insertion of a source may cause the variation of the velocity of soliton. This approach can be applied to all other (1(plus sign)1)-dimensional soliton hierarchies. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533420
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  • 5
    Electronic Resource
    Electronic Resource
    Families of dynamical r-matrices and Jacobi inversion problem for nonlinear evolution equations (1998)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 39 (1998), S. 5964-5983 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Three families of dynamical r-matrices are explicitly constructed by means of constrained flows of nonlinear evolution equations (NLEE) associated with polynomial second-order spectral problems. The dynamical classical Yang–Baxter equations for the three linear r-matrix algebras are presented with explicit expression for the extra term. The separation variables and the Jacobi inversion problems for the x- and tn-constrained flow of the NLEEs are found. The factorization of NLEE allows us to obtain the Jacobi inversion problem for the NLEE by combining the above two Jacobi inversion problems. This provides a method of separation of variables to solve the NLEEs. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532608
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  • 6
    Electronic Resource
    Electronic Resource
    Separation of variables for the constrained flows (1997)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 321-329 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The separation of variables for some constrained flows of soliton hierarchies are shown. The Lax matrix and r-matrix (either constant or dynamical type) are used to construct the separation variables and separation equation for the constrained flows. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531851
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  • 7
    Electronic Resource
    Electronic Resource
    Two binary Darboux transformations for the KdV hierarchy with self-consistent sources (2001)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 42 (2001), S. 2113-2128 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Two binary (integral type) Darboux transformations for the KdV hierarchy with self-consistent sources are proposed. In contrast with the Darboux transformation for the KdV hierarchy, one of the two binary Darboux transformations provides non-auto-Bäcklund transformation between two nth KdV equations with self-consistent sources with different degrees. The formula for the m-times repeated binary Darboux transformations are presented. This enables us to construct the N-soliton solution for the KdV hierarchy with self-consistent sources. © 2001 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1357826
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  • 8
    Electronic Resource
    Electronic Resource
    The bi-Hamiltonian structure for the restricted flows of the Boussinesq and the modified Boussinesq equation (1993)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 34 (1993), S. 4742-4754 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The methods of previous articles are extended herein, using the Miura-type transformation between two soliton equations and the gauge transformation between two associated eigenvalue problems, and a bi-Hamiltonian structure and involutive integrals of motion for x- and t-finite-dimensional integrable Hamiltonian systems (FDIHS) obtained from the decomposition of the Boussinesq equation and the modified Boussinesq equation are computed, respectively. This decomposition provides a way to obtain a certain kind of solution for the later two equations through solving two commuting FDIHS's. The methods are also applied to the FDIHS's related to the higher-order constraints.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530369
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  • 9
    Electronic Resource
    Electronic Resource
    The constraints of potentials and the finite-dimensional integrable systems (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 1679-1689 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Restricting potential to the space spanned by the eigenvectors of the recursion operator leads to a natural constraint of potential and a finite-dimensional integrable Hamiltonian system. The general method for proving the consistency of the two systems stemming from the Lax pair and obtaining the constants of the motion for the Hamiltonian system is illustrated by the classical Boussinesq and AKNS hierarchies. By using gauge transformation, similar results for the Jaulent–Miodek and Kaup–Newell hierarchies are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528253
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  • 10
    Electronic Resource
    Electronic Resource
    Families of quasi-bi-Hamiltonian systems and separability (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 4452-4473 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: It is shown how to construct an infinite number of families of quasi-bi-Hamiltonian (QBH) systems by means of the constrained flows of soliton equations. Three explicit QBH structures are presented for the first three families of the constrained flows. The Nijenhuis coordinates defined by the Nijenhuis tensor for the corresponding families of QBH systems are proved to be exactly the same as the separated variables introduced by mean of the Lax matrices for the constrained flows. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532979
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