Quasi-periodic and periodic solutions for coupled nonlinear Schrödinger equations of Manakov type. (English) Zbl 0965.35157

Christiansen, P. L.; Eilbeck, J. C.; Enolskii, V. Z.; Kostov, N. A.

Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 456, No. 2001, 2263-2281 (2000).

The authors study the following coupled nonlinear Schrödinger equations (NLS): \[ \begin{cases} iU_t+U_{xx}+(\kappa UU^*+\chi VV^*)V=0,\cr iV_t+V_{xx}+(\chi UU^*+\rho VV^*)V=0.\end{cases} \] The case \(\kappa=\chi=\rho=1\) is the integrable Manakov system. Using the ansatz proposed by Porubov and Parker the solutions are sought in the form \[ U_k(t,x)=q_k(x)\exp\left(ia_kt+ic_k\int_{x_0}^x{dx\over q_i^2(x)}\right), \quad k=1,2, \] with \(U_1=U\), \(U_2=V\), which under substitution is reduced to the Hamiltonian equation on \(T^*{\mathbb R}^2={\mathbb R}^4(p,q)\) (with time \(x\)) with \[ H={|p|^2\over 2}+Q_4(q)-{a_1q_1^2+a_2q_2^2\over 2}+ {c_1^2q_1^{-2}+c_2^2q_2^{-2}\over 2}. \] Depending on the homogeneous fourth-order polynomial \(Q_4\) there are different integrable cases. The authors study the bi-quadric case \((q_1^2+q_2^2)^2\), corresponding to the integration in ellipsoidal coordinates, though the other cases can be treated in this approach as well.
For this case they construct Lax representation and reduce the problem to the Jacobi inversion problem associated with a genus-two algebraic curve. Then the system is integrated in terms of Kleinian genus-two hyperelliptic functions. This allows to obtain new periodic solutions via reduction of genus-two hyperelliptic functions to (genus-one) elliptic functions and quasi-periodic solutions via spectral theory of NLS with elliptic potential.
The coupled NLS considered in the paper are important for a number of physical applications, as they describe propagation and transmission in fiber optics.

Reviewer: Boris S.Kruglikov (Tromsoe)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
37K20	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
35B15	Almost and pseudo-almost periodic solutions to PDEs
78A60	Lasers, masers, optical bistability, nonlinear optics

Keywords:

nonlinear Schrödinger equations; Manakov systems; Kleinian functions; algebraic curve; hyperelliptic functions; elliptic functions