The interaction ofn-dimensional soliton wave fronts

Abstract

We consider solutions of the equation $$\psi $$ , where $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} = \alpha ^{ij} \partial _i \partial _j (i,j = 0,...,n)$$ and where α is any constant symmetric matrix. The solutions behave as soliton-shaped wave fronts inn dimensions. These interact but keep their identity as in one dimension. We give solutions for the interaction of up to four of these waves for any matrix α and suggest a solution forN of them. There is only one essential difference between these solutions and those of the one-dimensional system;i.e. conditions are found which restrict the motion of then-dimensional waves and it is these conditions we find the most interesting. If we choose α=diag (−1, 1, 1), then it is found that any three interacting waves intersect and form a triangle. For this α we find that the area of this triangle must remain constant in time and hence the wave fronts are forced to move in a specific fixed configuration. We also briefly consider a type of two-dimensional Korteweg-de Vries equation which, although artificially constructed from the one-dimensional case, has similar solutions to the 2-dimensional sine-Gordon equation.