Тhe energy balance of the fixed solar plasma in magnetic tube, taking into account the longitudinal heat conduction, heating and radiative cooling is examined. The problem is reduced to the solution of nonlinear oscillator with potential. The maximum heat flux determines the depth of the potential well.
Potentials with the type of fourth-degree polynomial with two multiple roots corresponds to the solution of the transition region (TR). The expression for the generalized source, including heating and radiative cooling, has the form of alternating third-order polynomial. The intermediate root is determined by the condition of stability.
It is shown that the evolution of TR in the hot phase of the outbreak is accompanied by an increase in the depth of the potential well, increasing the maximum temperature and the movement of TR to the photosphere. These findings are consistent with semi-empirical models and numerical calculations.
When the transformation capacity, which resulted in broken multiplicity two roots, corresponding to the maximum temperature in a closed magnetic tube temperature distribution is a soliton, localized at the apex. It is assumed that the observed localized X-ray sources in the tops of magnetic tubes are thermal solitons. If the adjustment capacity of all four roots are unequal, there is a solution corresponding to the periodic standing wave.