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

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.

### MSC:

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

  Adams, R., Linear $$q$$, Bull. Amer. Math. Soc., 37, 361-400 (1931) · JFM 57.0534.05  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  Arik, M.; Coon, D., Hilbert spaces of analytic functions and generalized coherent states, J. Math. Phys., 17, 524-527 (1976) · Zbl 0941.81549  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)  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  Degasperis, A.; Shabat, A., Construction of reflectionless potentials with infinite discrete spectrum, Theoret. and Math. Phys., 100, 970-984 (1994) · Zbl 0857.34073  Derfel, G. A.; Iserles, A., The pantograph equation in the complex plane, J. Math. Anal. Appl., 213, 117-132 (1997) · Zbl 0891.34072  Feldstein, A.; Liu, Y., On neutral functional differential equations with variable delays, Proc. Cambridge Philos. Soc., 124, 371-384 (1998) · Zbl 0913.34067  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  Gasper, G.; Rahman, M., Basic Hypergeometric Series (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0695.33001  Gaver, D. P., An absorption probability problem, J. Math. Anal. Appl., 9, 384-393 (1964) · Zbl 0235.60108  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  Gripenberg, G.; Londen, S.-O.; Staffans, O., Volterra Integral and Functional Equations (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0695.45002  Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1977), Springer-Verlag: Springer-Verlag New York  Infeld, L.; Hull, T. E., Factorization method, Rev. Modern Phys., 23, 21-68 (1951) · Zbl 0043.38602  Iserles, A., On the generalized pantograph functional-differential equation, European J. Appl. Math., 4, 1-38 (1992) · Zbl 0767.34054  Iserles, A., On nonlinear delay differential equations, Trans. Amer. Math. Soc., 344, 441-477 (1994) · Zbl 0804.34065  Iserles, A.; Terjéki, J., Stability and asymptotic stability of functional-differential equations, J. London Math. Soc. (2), 51, 559-572 (1995) · Zbl 0832.34080  Iserles, A.; Liu, Y., On functional-differential equations with proportional delays, J. Math. Anal. Appl., 207, 73-95 (1997) · Zbl 0873.34066  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  Liu, Y., Stability of $$θ$$, Numer. Math., 70, 473-485 (1995) · Zbl 0824.65081  Liu, Y., Asymptotic behavior of functional-differential equations with proportional time delays, European J. Appl. Math., 7, 11-30 (1996) · Zbl 0856.34078  Macfarlane, A. J., On $$q_q$$, J. Phys. A, 22, 4581-4588 (1989) · Zbl 0722.17009  Nussbaum, R., Existence and uniqueness theorems for some functional differential equations of neutral type, J. Differential Equations, 11, 607-623 (1972) · Zbl 0263.34070  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)  Shabat, A., The infinite dimensional dressing dynamical system, Inverse Problems, 8, 303-308 (1992) · Zbl 0762.35098  Skorik, S.; Spiridonov, V., Self-similar potentials and the $$q$$, Lett. Math. Phys., 28, 59-74 (1993) · Zbl 0778.17012  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  Spiridonov, V., Exactly solvable potentials and quantum algebras, Phys. Rev. Lett., 69, 398-401 (1992) · Zbl 0968.81524  Spiridonov, V., Universal superpositions of coherent states and self-similar potentials, Phys. Rev. A, 52, 1909-1935 (1995)  Spiridonov, V., Coherent states of the $$q$$, Lett. Math. Phys., 35, 179-185 (1995) · Zbl 0833.17013  Spiridonov, V.; Zhedanov, A., Symmetry preserving quantization and self-similar potentials, J. Phys. A, 28, L589-L595 (1995)  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  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
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