Stable three-dimensional modon soliton in plasmas

We derive the nonlinear equations that describe coupled drift waves and ion acoustic waves in a plasma. We show that when the coupling to ion acoustic waves is negligible, the reduced nonlinear equation is a generalization of the Hasegawa-Mima equation to the three-dimensional (3D) case. We find an exact analytical solution of this equation in the form of a 3D soliton drift wave (3D modon). By numerical simulations we study collisions between the modons and show that the collisions can be fully elastic.

Figure 1

(Top row) The function $\mathrm{\Psi }$ as a function of the radial coordinate for the three lowest states with $n=1,2,3$. The modon parameters are $v_d=0.1, a=1, u=1., p=2$. (Bottom row) The same for the function ${\mathrm{\Psi }}_0$.

Figure 2

Elastic head-on collision between the modons. The modon parameters are $v_1=0.3, v_2=-0.5, a_1=0.5, a_2=0.5, p_1=p_2={\mu }_1={\mu }_2=0$. (Left column) The field distribution $\mathrm{\Phi }$ in the $x-y$ plane; (right column) the isosurface $\mathrm{\Phi }(x,y,z)=0.15$.

Figure 3

Elastic overtaking collision between the modons. The modon parameters are $v_1=-0.3, v_2=-0.01, a_1=0.5, a_2=2, p_1=p_2={\mu }_1={\mu }_2=0$. (Left column) The field distribution $\mathrm{\Phi }$ in the $x-y$ plane; (right column) isosurface $\mathrm{\Phi }(x,y,z)=0.08$.

Figure 4

Inelastic head-on collision of the modons with radially symmetric part. The modon parameters are $v_1=0.3, v_2=-0.5, a_1=0.5, a_2=0.5, p_1=2., p_2=0.3, {\mu }_1={\mu }_2=0$. (Left column) Contour plots of $\mathrm{\Phi }$ in the $x-y$ plane; (right column) isosurfaces $\mathrm{\Phi }(x,y,z)=0.2$.

Figure 5

Inelastic head-on collision of the modons with $z$-antisymmetric part. The amplitudes of modons after the collision are changed due to emitted radiation. The modon parameters are $v_1=0.3, v_2=-0.5, a_1=0.5, a_2=0.5, p_1=p_2=0, {\mu }_1=1., {\mu }_2=0.5$. (Left column) Contour plots of $\mathrm{\Phi }$ in the $y-z$ plane; (right column) isosurfaces $\mathrm{\Phi }(x,y,z)=0.2$.