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June  2021, 29(2): 2141-2147. doi: 10.3934/era.2020109

Integrating evolution equations using Fredholm determinants

Feride Tığlay ,

Ohio State University, Newark, 1179 University Drive, Newark, OH 43055, USA

* Corresponding author: Feride Tığlay

Received  January 2020 Revised  August 2020 Published  June 2021 Early access  October 2020

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.

Keywords: Evolution equations, Fredholm determinants, integrability, Hunter-Saxton equation.
Mathematics Subject Classification: Primary:35M13, 35C15;Secondary:37K10.
Citation: Feride Tığlay. Integrating evolution equations using Fredholm determinants. Electronic Research Archive, 2021, 29 (2) : 2141-2147. doi: 10.3934/era.2020109
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.  doi: 10.1137/0151075.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

S. Lang, Differential Manifolds, Second edition. Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4684-0265-0.  Google Scholar

[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.  Google Scholar

[15]

H. P. McKean, Fredholm determinants and the Camassa-Holm hierarchy, Comm. Pure Appl. Math., 56 (2003), 638-680.  doi: 10.1002/cpa.10069.  Google Scholar

[16]

H. P. McKean, Breakdown of the Camassa-Holm equation, Comm. Pure Appl. Math., 57 (2004), 416-418.  doi: 10.1002/cpa.20003.  Google Scholar

[17]

M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser Boston, Inc., Boston, MA, 1991. doi: 10.1007/978-1-4612-0431-2.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.  doi: 10.1137/0151075.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

S. Lang, Differential Manifolds, Second edition. Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4684-0265-0.  Google Scholar

[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.  Google Scholar

[15]

H. P. McKean, Fredholm determinants and the Camassa-Holm hierarchy, Comm. Pure Appl. Math., 56 (2003), 638-680.  doi: 10.1002/cpa.10069.  Google Scholar

[16]

H. P. McKean, Breakdown of the Camassa-Holm equation, Comm. Pure Appl. Math., 57 (2004), 416-418.  doi: 10.1002/cpa.20003.  Google Scholar

[17]

M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser Boston, Inc., Boston, MA, 1991. doi: 10.1007/978-1-4612-0431-2.  Google Scholar

[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.  Google Scholar

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