Problem of stability of multidimensional solutions of the BK class equations in space plasma

The problem of stability of the multidimensional solutions of the BK class equations describing the nonlinear waves which are forming on the low-frequency branch of oscillations in plasma for cases when β≡4πnT/B2≪1 and β>1 is studied. In first case, for ω<ωB=eB/Mc,kλD≪1 the FMS waves are excited, and their dynamics under conditions kx2≫k⊥2, vx≪cA near the cone of θ=arctan(M/m)1/2, is described by the equation of the BK class known as the GKP equation for magnetic field h=B∼/B with due account of the high order dispersive correction defined by values of plasma parameters and angle θ=(B,k). In another case, the dynamics of the finite-amplitude Alfvén waves propagating near-to-parallel to B is described by the equation of the same class known as the 3-DNLS equation for h=(By+iBz)/2B|1-β|. To study the stability of multidimensional solutions in both cases the method of investigation of the Hamiltonian bounding with deformation conserving momentum by solving the variation problem is used. As a result, we have obtained the conditions of existence of the 2D and 3D soliton solutions in the BK system for cases of the GKP and 3-DNLS equation (i.e. for the FMS and Alfvén waves, respectively) in dependence on the equations' coefficients, i.e. on the parameters of both plasma and wave.