Josephson Junctions with Phase Shifts Stability Analysis of Fractional Fluxons

Dit proefscrift werd mede mogelijk gemaakt met financï ele steun van de Koninklijke Nederlandse Akademie van Wetenschappen (KNAW). " Allah will exalt those who believe among you and those who have knowledge to high ranks. " (QS Al-Mujaadalah 58:11) for: my beloved ones Contents 1 Josephson junctions and sine-Gordon equations: an Introduction 3 1.1 Josephson junctions with phase shift 5 1.2 Sine-Gordon equations 11 1.3 Outline of the thesis 13 Bibliography 15 2 Static semifluxons of a Josephson junction with π-discontinuity points 19 2.1 Introduction 21 2.2 Mathematical model and phase-plane analysis 22 2.3 Junctions with a single π-discontinuity point 23 2.4 Junctions with two π-discontinuity points 27 2.4.1 Antisemifluxon-semifluxon case 31 2.4.2 Semifluxon-antisemifluxon case 32 2.5 Junctions with multiple π-discontinuity points 33 2.6 Conclusions 37 Bibliography 39 3 Stability analysis of solitary waves in a 0-π Josephson junction 41 3.1 Introduction 43 3.2 Mathematical equation and its interpretation as junction model 44 3.2.1 Discrete 0-π sine-Gordon equation 44 3.2.2 Various approximation to the discreteness in the continuum limit 46 3.3 π-kink and its spectra in the continuum limit 48 3.3.1 Stability of the type 1 solution 52 3.3.2 Instability of type 2 solutions 56 3.3.3 Instability of type 3 solutions 59 3.4 Lattice π-kinks and their spectra in the discrete case: continuum approximation 64 3.4.1 Stability of type 1 semifluxon 66 3.4.2 Instability of type 2 semifluxon 68 3.4.3 Instability of type 3 semifluxon 69 3.5 Semikinks in the weak-coupling limit 70 3.6 Numerical computations of the discrete system 74 3.6.1 Stability of type 1 lattice soliton 75 3.6.2 Instability of type 2 lattice soliton 77 3.6.3 Instability of type 3 lattice solution 80 3.7 Conclusions 80 Bibliography 83 4 Stability analysis of solitary waves in a tricrystal junction 87