Regular solutions of the Shabat equation. (English) Zbl 0929.34054

Liu, Yunkang

J. Differ. Equations 154, No. 1, 1-41 (1999).

The author discusses the Shabat equation \[ f'(t)+q^2f'(qt)+f^2(t)-q^2f^2(qt)=\mu,\tag{1} \] where \(q\) and \(\mu\) are real parameters, \(q\in (-1,0)\cup (0,1)\). A function \(f(t)\) is called a global solution to (1) if \(f(t)\in C^1(\mathbb{R})\) and it satisfies (1) for every \(t\in\mathbb{R}\). In the case \(\mu >0\), a function \(f(t)\) is called a regular solution to (1) if it is a global solution to (1) and it satisfies the asymptotic expansions \[ f(t)=\pm f^*+O(t^{-2})\qquad\text{and}\qquad f'(t)=O(t^{-3})\quad\text{as }t\to\pm\infty,\text{ with }f^*=\sqrt{\mu\big /(1-q^2)}. \] Considering the initial condition (2) \(f(0)=f_0\) the author proves that the initial value problem (1), (2) has a unique solution in a neighborhood of the origin and set up conditions under which the solution is regular, resp. it is not global. It is shown that for \(\mu >0\) any global solution \(f(t)\) to (1) satisfying \(f(t)\to\pm f^*\) as \(t\to\pm\infty\) is a regular solution.
The author discusses also some related topics, including \(q\)-difference equations, neutral equations and Dirichlet series.

Reviewer: Jan Čermák (Brno)

MSC:

34K12	Growth, boundedness, comparison of solutions to functional-differential equations
34K05	General theory of functional-differential equations
34K40	Neutral functional-differential equations

Keywords:

Shabat equation; global solution; regular solution; asymptotic behavior

References:

[1]	Adams, R., Linear \(q\), Bull. Amer. Math. Soc., 37, 361-400 (1931) · JFM 57.0534.05
[2]	Adler, V. E., On the rational solutions of the Shabat equations, Nonlinear Physics: Theory and Experiment (1996), World Scientific: World Scientific Singapore, p. 3-10 · Zbl 0941.34515
[3]	Arik, M.; Coon, D., Hilbert spaces of analytic functions and generalized coherent states, J. Math. Phys., 17, 524-527 (1976) · Zbl 0941.81549
[4]	Barclay, D. T.; Dutt, R.; Gangopadhyaya, A.; Khare, A.; Pagnamenta, A.; Sukhatme, U., New exactly solvable Hamiltonian: Shape invariance and self-similarity, Phys. Rev. A, 48, 2786-2797 (1993)
[5]	Carr, J.; Dyson, J., The functional differential equation \(yx ay λx by x\), Proc. Roy. Soc. Edinburgh Sect. 7, 4A, 165-174 (1974/75) · Zbl 0344.34059
[6]	Degasperis, A.; Shabat, A., Construction of reflectionless potentials with infinite discrete spectrum, Theoret. and Math. Phys., 100, 970-984 (1994) · Zbl 0857.34073
[7]	Derfel, G. A.; Iserles, A., The pantograph equation in the complex plane, J. Math. Anal. Appl., 213, 117-132 (1997) · Zbl 0891.34072
[8]	Feldstein, A.; Liu, Y., On neutral functional differential equations with variable delays, Proc. Cambridge Philos. Soc., 124, 371-384 (1998) · Zbl 0913.34067
[9]	Frederickson, P. O., Dirichlet series solution for certain functional differential equations, Japan-United States Seminar on Ordinary Differential and Functional Equations (1971), Springer-Verlag, p. 247-254 · Zbl 0191.15302
[10]	Gasper, G.; Rahman, M., Basic Hypergeometric Series (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0695.33001
[11]	Gaver, D. P., An absorption probability problem, J. Math. Anal. Appl., 9, 384-393 (1964) · Zbl 0235.60108
[12]	Gelfand, I. M.; Fairlie, D. B., The algebra of Weyl symmetrized polynomials and its quantum extensions, Comm. Math. Phys., 136, 487-499 (1991) · Zbl 0735.17029
[13]	Gripenberg, G.; Londen, S.-O.; Staffans, O., Volterra Integral and Functional Equations (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0695.45002
[14]	Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1977), Springer-Verlag: Springer-Verlag New York
[15]	Infeld, L.; Hull, T. E., Factorization method, Rev. Modern Phys., 23, 21-68 (1951) · Zbl 0043.38602
[16]	Iserles, A., On the generalized pantograph functional-differential equation, European J. Appl. Math., 4, 1-38 (1992) · Zbl 0767.34054
[17]	Iserles, A., On nonlinear delay differential equations, Trans. Amer. Math. Soc., 344, 441-477 (1994) · Zbl 0804.34065
[18]	Iserles, A.; Terjéki, J., Stability and asymptotic stability of functional-differential equations, J. London Math. Soc. (2), 51, 559-572 (1995) · Zbl 0832.34080
[19]	Iserles, A.; Liu, Y., On functional-differential equations with proportional delays, J. Math. Anal. Appl., 207, 73-95 (1997) · Zbl 0873.34066
[20]	Kato, T.; McLeod, J. B., The functional-differential equation \(yx ay λx by x\), Bull. Amer. Math. Soc., 77, 891-937 (1971) · Zbl 0236.34064
[21]	Liu, Y., Stability of \(θ\), Numer. Math., 70, 473-485 (1995) · Zbl 0824.65081
[22]	Liu, Y., Asymptotic behavior of functional-differential equations with proportional time delays, European J. Appl. Math., 7, 11-30 (1996) · Zbl 0856.34078
[23]	Macfarlane, A. J., On \(q_q\), J. Phys. A, 22, 4581-4588 (1989) · Zbl 0722.17009
[24]	Nussbaum, R., Existence and uniqueness theorems for some functional differential equations of neutral type, J. Differential Equations, 11, 607-623 (1972) · Zbl 0263.34070
[25]	Ockendon, J. R.; Tayler, A. B., The dynamics of a current collection system for an electric locomotive, Proc. Roy. Soc. Edinburgh Sect. A, 322, 447-468 (1971)
[26]	Shabat, A., The infinite dimensional dressing dynamical system, Inverse Problems, 8, 303-308 (1992) · Zbl 0762.35098
[27]	Skorik, S.; Spiridonov, V., Self-similar potentials and the \(q\), Lett. Math. Phys., 28, 59-74 (1993) · Zbl 0778.17012
[28]	Spiridonov, V., Nonlinear algebras and spectral problems, (Le Tourneux, J.; Vinet, L., Quantum Groups, Integral Models and Statistical Systems (1992), World Scientific: World Scientific Singapore), 246-256
[29]	Spiridonov, V., Exactly solvable potentials and quantum algebras, Phys. Rev. Lett., 69, 398-401 (1992) · Zbl 0968.81524
[30]	Spiridonov, V., Universal superpositions of coherent states and self-similar potentials, Phys. Rev. A, 52, 1909-1935 (1995)
[31]	Spiridonov, V., Coherent states of the \(q\), Lett. Math. Phys., 35, 179-185 (1995) · Zbl 0833.17013
[32]	Spiridonov, V.; Zhedanov, A., Symmetry preserving quantization and self-similar potentials, J. Phys. A, 28, L589-L595 (1995)
[33]	Staffans, O., A neutral FDE is similar to the product of an ODE and a shift, J. Math. Anal. Appl., 192, 627-654 (1995) · Zbl 0833.34058
[34]	Veselov, A. P.; Shabat, A. B., The dressing dynamical system and spectra theory of Schrödinger operator, Funct. Anal. Appl., 27, 81-96 (1993) · Zbl 0813.35099