Generalization of an integro-transcendetnal equation system with vanishing solutions in free phase problem

On the Mutual Transformation of the Types of Solutions of one Class of Nonlinear Hammerstein Integral Equations

“…In the present work, we perform the theoretical investigation of the cases in which the required relationship between the indicated two sets of solutions is possible and in what cases these sets cannot have common points, although the distance between them can be very small (such that it is difficult to detect it numerically). This situation was observed in one of examples considered in [7].…”

“…Hence, in this case, the system of equations (10) cannot be reduced to system (14). Therefore, for the indicated given function, a solution of type (3) cannot transform into a solution of type (7). System (10) contains 2N equations for the determination of 2N real unknowns ( N real and imaginary parts of the parameters η Nk ).…”

“…In the works [4,5], one can find some numerical results obtained for this problem with t = 0.1. In this case, within a certain range of the parameter c , the values of the imaginary part η 11 of the solution passing through zero are close to zero.…”

“…The problem of mutual transformation of solutions of different types was numerically studied in [7]. The numerical results obtained in the cited work enable us to assume that it is possible to transform a complex solution generated by a polynomial of the first degree that does not have zeros in the domain of definition into a real solution that passes through zero.…”

“…The works [4][5][6][7] are devoted to the application of the concept of generating polynomials for the description of solutions. The numerical results obtained earlier show that the solutions of both types may exist at the same time.…”