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Integrating evolution equations using Fredholm determinants
Ohio State University, Newark, 1179 University Drive, Newark, OH 43055, USA |
We outline the construction of special functions in terms of Fredholm determinants to solve boundary value problems of the string spectral problem. Our motivation is that the string spectral problem is related to the spectral equations in Lax pairs of at least three nonlinear evolution equations from mathematical physics.
References:
[1] |
V. Arnold,
Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses application à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361.
doi: 10.5802/aif.233. |
[2] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[3] |
K. Colville, D. Gomez and J. Szmigielski,
On isospectral deformations of an inhomogeneous string, Comm. Math. Phys., 348 (2016), 771-802.
doi: 10.1007/s00220-016-2711-y. |
[4] |
R. E. Eaves,
A sufficient condition for the convergence of an infinite determinant, SIAM J. Appl. Math., 18 (1970), 652-657.
doi: 10.1137/0118058. |
[5] |
B. Fuchssteiner and A. S. Fokas,
Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[6] |
J. K. Hunter and R. Saxton,
Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.
doi: 10.1137/0151075. |
[7] |
I. S. Kac and G. Krein, On the spectral functions of the string, Amer. Math. Soc. Transl., 103 (1974), 19-102. Google Scholar |
[8] |
B. Khesin, J. Lenells and G. Misiołek,
Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann., 342 (2008), 617-656.
doi: 10.1007/s00208-008-0250-3. |
[9] |
B. Khesin and G. Misiołek,
Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176 (2003), 116-144.
doi: 10.1016/S0001-8708(02)00063-4. |
[10] |
B. Khesin and R. Wendt, The Geometry of Infinite-Dimensional Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 51, Springer-Verlag, Berlin, 2009. |
[11] |
A. A. Kirillov, Infinite-dimensional Lie groups: Their orbits, invariants and representations. The geometry of moments, Lect. Notes in Math., Springer-Verlag, New York, 970 (1982), 101–123.
doi: 10.1007/BFb0066026. |
[12] |
A. A. Kirillov and D. V. Yuriev,
Kähler geometry of the infinite-dimensional homogeneous space $M = {\rm{Diff}}_+ (S^1)/{\rm{Rot}}(S^1)$, Funktsional. Anal. i Prilozhen., 21 (1987), 35-46.
|
[13] |
S. Lang, Differential Manifolds, Second edition. Springer-Verlag, New York, 1985.
doi: 10.1007/978-1-4684-0265-0. |
[14] |
J. Lenells, G. Misiołek and F. Tiğlay,
Integrable evolution equations on spaces of tensor densities and their peakon solutions, Comm. Math. Phys., 299 (2010), 129-161.
doi: 10.1007/s00220-010-1069-9. |
[15] |
H. P. McKean,
Fredholm determinants and the Camassa-Holm hierarchy, Comm. Pure Appl. Math., 56 (2003), 638-680.
doi: 10.1002/cpa.10069. |
[16] |
H. P. McKean,
Breakdown of the Camassa-Holm equation, Comm. Pure Appl. Math., 57 (2004), 416-418.
doi: 10.1002/cpa.20003. |
[17] |
M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser Boston, Inc., Boston, MA, 1991.
doi: 10.1007/978-1-4612-0431-2. |
[18] |
F. Tiğlay and C. Vizman,
Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications, Lett. Math. Phys., 97 (2011), 45-60.
doi: 10.1007/s11005-011-0464-2. |
show all references
References:
[1] |
V. Arnold,
Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses application à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361.
doi: 10.5802/aif.233. |
[2] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[3] |
K. Colville, D. Gomez and J. Szmigielski,
On isospectral deformations of an inhomogeneous string, Comm. Math. Phys., 348 (2016), 771-802.
doi: 10.1007/s00220-016-2711-y. |
[4] |
R. E. Eaves,
A sufficient condition for the convergence of an infinite determinant, SIAM J. Appl. Math., 18 (1970), 652-657.
doi: 10.1137/0118058. |
[5] |
B. Fuchssteiner and A. S. Fokas,
Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[6] |
J. K. Hunter and R. Saxton,
Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.
doi: 10.1137/0151075. |
[7] |
I. S. Kac and G. Krein, On the spectral functions of the string, Amer. Math. Soc. Transl., 103 (1974), 19-102. Google Scholar |
[8] |
B. Khesin, J. Lenells and G. Misiołek,
Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann., 342 (2008), 617-656.
doi: 10.1007/s00208-008-0250-3. |
[9] |
B. Khesin and G. Misiołek,
Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176 (2003), 116-144.
doi: 10.1016/S0001-8708(02)00063-4. |
[10] |
B. Khesin and R. Wendt, The Geometry of Infinite-Dimensional Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 51, Springer-Verlag, Berlin, 2009. |
[11] |
A. A. Kirillov, Infinite-dimensional Lie groups: Their orbits, invariants and representations. The geometry of moments, Lect. Notes in Math., Springer-Verlag, New York, 970 (1982), 101–123.
doi: 10.1007/BFb0066026. |
[12] |
A. A. Kirillov and D. V. Yuriev,
Kähler geometry of the infinite-dimensional homogeneous space $M = {\rm{Diff}}_+ (S^1)/{\rm{Rot}}(S^1)$, Funktsional. Anal. i Prilozhen., 21 (1987), 35-46.
|
[13] |
S. Lang, Differential Manifolds, Second edition. Springer-Verlag, New York, 1985.
doi: 10.1007/978-1-4684-0265-0. |
[14] |
J. Lenells, G. Misiołek and F. Tiğlay,
Integrable evolution equations on spaces of tensor densities and their peakon solutions, Comm. Math. Phys., 299 (2010), 129-161.
doi: 10.1007/s00220-010-1069-9. |
[15] |
H. P. McKean,
Fredholm determinants and the Camassa-Holm hierarchy, Comm. Pure Appl. Math., 56 (2003), 638-680.
doi: 10.1002/cpa.10069. |
[16] |
H. P. McKean,
Breakdown of the Camassa-Holm equation, Comm. Pure Appl. Math., 57 (2004), 416-418.
doi: 10.1002/cpa.20003. |
[17] |
M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser Boston, Inc., Boston, MA, 1991.
doi: 10.1007/978-1-4612-0431-2. |
[18] |
F. Tiğlay and C. Vizman,
Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications, Lett. Math. Phys., 97 (2011), 45-60.
doi: 10.1007/s11005-011-0464-2. |
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