In the literature, any eventually nontrivial solution
x [member of] [M.
It's well known that every nontrivial solution
of (1) is entire and of infinite order.
Due to the special structure of the nonlinear Schrodinger equation, the Jacobian operator exhibits one eigenvalue that moves to zero when the Newton iterate converges to a nontrivial solution
and is exactly zero at a solution.
One nontrivial solution
is constructed using the classical mountain pass theorem.
For nontrivial solution
the determinant of the coefficient matrix [W([alpha], [xi], [[eta].
r] for which Equation (2) has a nontrivial solution
The vanishing determinant of a nontrivial solution
Next, using Morse theory, we will produce a third nontrivial solution
for problem (1.
For a nontrivial solution
of (18), the determinant of coefficient matrix should be zero.
A complex frequency is introduced in order to find a nontrivial solution
of characteristic Equations [9,10] that permits then to find the surface current (more precisely the coefficients of the basis functions expansion used to approximate the surface current).
Let the functions r(t) > 0 and u(t) exist for every nontrivial solution
x(t), t [member of] [J.
A nontrivial solution
of system (8) exists if the characteristic equation ([[alpha].