EXISTENCE OF ASYMPTOTIC SOLUTIONS TO SEMILINEAR PARTIAL DIFFERENCE EQUATIONS ON GRAPHS A RESEARCH EXPERIENCE FOR UNDERGRADUATES

Abstract. This paper studies nonlinear partial difference equations on graphs. We seek solutions to the semilinear equation −Lu + su + u3 = 0 where L is the Laplacian of a graph G = (V, E). In particular we prove the existence of 3n solutions when s → − ∞ and n = |V |. In addition, we find their Morse indices and exact forms. In [4], the authors used the tGNGA method to produce bifurcation diagrams for several graphs; however, these diagrams are not complete. This study complements [4] by using the as-ymptotic solutions to construct a complete bifurcation diagram. 1