Nonlinear interactions of tenuous electron beam, background, unmagnetized plasma, and self-consistently generated Langmuir and ion-sound waves are analyzed in the framework of plasma weak turbulence kinetic theory. Full numerical solutions of the complete weak turbulence equations are obtained for the first time, which show the familiar plateau formation in the electron beam distribution and concomitant quasi-saturation of primary Langmuir waves, followed by fully nonlinear processes which include three-wave decay and induced-scattering processes. A detailed analysis reveals that the scattering off ions is an important nonlinear process which leads to prominent backscattered and long-wavelength Langmuir wave components. However, it is found that the decay process is also important, and that the nonlinear development of weak Langmuir turbulence critically depends on the initial conditions. Special attention is paid to the electron-to-ion temperature ratio, Te/Ti, and the initial perturbation level. It is found that higher values of Te/Ti promote the generation of backscattered Langmuir wave component, and that a higher initial wave intensity suppresses the backscattered component while significantly enhancing the long-wavelength Langmuir wave component.