Quasi-periodic and periodic solutions for coupled nonlinear Schrödinger equations of Manakov type (Q2706728)

The authors study the following coupled nonlinear Schrödinger equations (NLS): NEWLINE\[NEWLINE \begin{cases} iU_t+U_{xx}+(\kappa UU^*+\chi VV^*)V=0,\cr iV_t+V_{xx}+(\chi UU^*+\rho VV^*)V=0.\end{cases} NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE U_k(t,x)=q_k(x)\exp\left(ia_kt+ic_k\int_{x_0}^x{dx\over q_i^2(x)}\right), \quad k=1,2, NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE H={|p|^2\over 2}+Q_4(q)-{a_1q_1^2+a_2q_2^2\over 2}+ {c_1^2q_1^{-2}+c_2^2q_2^{-2}\over 2}. NEWLINE\]NEWLINE 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. NEWLINENEWLINENEWLINEFor 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. NEWLINENEWLINENEWLINEThe coupled NLS considered in the paper are important for a number of physical applications, as they describe propagation and transmission in fiber optics.