In this chapter we explain the resultson effective dynamics of solitons in slowly varying external potentials. We also present a list of further generalizations due to J. Fröhlich, S. Gustafson, B. L. G. Jonsson, I. M. Sigal, D. Stuart, T.-P. Tsai, H.-T. Yau, and others. Moreover, we comment on the relation to the mass--energy equivalence of Einstein.

We explain in detail the strategy of Buslaev--Perelman--Sulem (BPS): symplectic projection of a trajectory on a solitary manifold andmodulation equations for the projection and time decay for a transversal component using the Poincar\'e normal form and Fermi Golden Rule for the transversal dynamics. We present an extensive list of results onasymptotic stability of stationary orbitsand solitons that rely on the BPS strategy and its generalizations byS. Cuccagna, Y. Martel, F. Merle, T. Mizumachi, K. Nakanishi, I. Rodnianski,W. Schlag,I. M. Sigal, A. Soffer,R. L. Pego,T. P. Tsai,M. I. Weinstein, H. T. Yau, and others. We also mention the results on stability and instability ofself-similar, spherically symmetric solutions and rotating Kerr solutions ofequations of the General Theory of Relativity by T. Harada, C. E. Kenig, H. Maeda,F. Merle, W. Schlag, and others. Moreover, we illustrate the BPS strategyin the simplest modelof a 1D Schrödinger equation coupled to a nonlinear oscillator, giving complete proofs with all details.

In this chapter we describe the results of joint work withArkady Vinnichenko (1945--2009) on numerical simulation of (1) global attraction to solitonsand (2) adiabatic effective dynamics of the solitons forrelativistic 1D nonlinear wave equations.