Fig. 1. (a) Three gaps in the spectrum of the linear operator (10) associated with the NLS equation. To accurately represent the spectrum, six gap edges r_i are required. These gap edges correspond to six parameters in a 2-phase solution Ψ₂(9) of the NLS equation. (b) A degenerate gap forms when two gap edges are equal r₂=r₃. The spectrum is now represented by four gap edges r₁, r₄, r₅, r₆, corresponding to four parameters in a 1-phase solution Ψ₁(6) of the NLS equation. (c) Now two gaps are degenerate, leaving only one gap and two gap edges r₁ and r₄, which correspond to parameters in a 0-phase solution Ψ₀(3) of the NLS equation.

Fig. 2. Error in numerical method for hyperelliptical integral evaluation as a function of N, the number of Chebyshev polynomials used in the expansion of H_j(t) in Eq. (18). The different error curves correspond to different values for gap separation r₃−r₂=ν in Eqs. (19).

Fig. 3. Initial data regularization for a single shock initial condition (ρ(x,0)=ρ₀ and when x<0; ρ(x,0)=1 and u(x,0)=0 for x>0). Overlapping of the initial data for pairs of Riemann invariants correspond to gap degeneracies hence “cancel out”.

Fig. 4. Initial data for two shock collision.

Fig. 5. Cases corresponding to different choices for the initial velocity u₀ as a function of the initial density ρ₀ in Eq. (25). There are five different regions that give qualitatively different results for the collision interaction. The long time, asymptotic state for each case is labelled. A summary of the collision process for each region is given in Section 5.1.7. A typical initial asymptotic solution for each region is shown in Fig. 6.

Fig. 6. Density ρ as a function of x in the asymptotic solution to the collision initial value problem before interaction occurs. Each labelled plot corresponds to a particular choice of ρ₀ and u₀ from a region of parameter space in Fig. 5. All plots assume ρ₀=3 and L=1. The behaviour of the solution at each initial step is described as follows (see also [19]): I, upper left, : two DSWs connected by a pure 1-phase periodic region whose boundaries are delineated by vertical lines. II & III, upper right, : two DSWs connected by a constant 0-phase region. Case IV, lower left, u₀=0: one rarefaction wave connected to a DSW by a constant 0-phase region. Case V, lower right, : two rarefaction waves connected by a constant 0-phase region. For the 1-phase results of cases I, II, and III, ε=0.075, whereas for case IV, ε=0.035. All the 0-phase results assume ε=0.

Fig. 7. Initial data for two pure DSWs propagating toward each other (r₋ and r₊, dashed) and their regularization (r_i, solid).

Fig. 8. Evolution of the 2-phase Riemann invariants. The time is shown in the upper left corner for each snapshot.

Fig. 9. Bifurcation diagram for a two DSW collision with initial velocity . Different phase regions corresponding to the number of phases in the slowly modulated wave solution are marked. The straight line segments separating 0- and 1-phase regions are calculated from 1-phase DSW theory. Their slopes are the inverses of the front speeds. The filled region bounded by arcs corresponding to boundaries between 1 and 2-phase regions is determined by solving the 2-phase Whitham equations (11).

Fig. 10. Numerical solution of the density ρ for the collision of two DSWs. Dashed, vertical lines correspond to theoretically determined boundaries between different phase regions (see Fig. 8 and Fig. 9). Note the 2-phase interaction region that develops and changes into a 0-phase constant region with two DSWs propagating away from each other. Parameters are ρ₀=3 and L=1.

Fig. 11. Numerical solution of the velocity u for the collision of two DSWs with the theoretically determined boundaries between different phase regions marked with vertical dashed lines. Parameters are ρ₀=3 and L=1.

Fig. 12. Contour plot of numerical solution with overlay of bifurcation diagram showing the accuracy of Whitham theory. See Fig. 13 for a close-up of the 2-phase interaction region inside the dashed box.

Fig. 13. Close-up of 2-phase interaction region from Fig. 12. The 2-phase region near the 1-phase boundaries (the soliton limit) is characterized by a hexagonal lattice pattern corresponding to interacting nonlinear waves.

Fig. 14. Initial data regularization for a two DSW collision process with zero initial velocity.

Fig. 15. Phase regions for a two DSW collision process with zero initial velocity, similar to the pure shock case depicted in Fig. 9. The dashed rays represent the boundaries of two 0-phase rarefaction waves.

Fig. 16. Evolution of the density ρ from numerical simulation of equations (2) for the case u₀=0 in (25). Note the generation of two rarefaction waves. Parameters are ρ₀=3 and L=1.

Fig. 17. Contour plot of numerical solution for a DSW collision with the bifurcation diagram in Fig. 15 overlayed on top. The 2-phase region is characterized by a hexagonal lattice pattern corresponding to nonlinear interacting waves [18].

Fig. 18. An example bifurcation diagram in the characteristic xt plane for a collision process with large initial velocity corresponding to regions I and II in Fig. 5. The 2-phase interaction region (filled) expands in time, different from the previous cases analysed. The curves represent edges of the calculated rarefaction wave solutions for the 2-phase Riemann invariants (see Fig. 20 for the solution at t=1.5). The parameters are L=1, ρ₀=3, and .

Fig. 19. Density ρ of the numerical solution to the NLS equation (1) with two step initial data (25) for large initial velocity at different times (noted on left of each plot). In this regime, the 2-phase interaction region remains for all time but is asymptotically degenerate. The parameters are L=1, ε=0.15, and ρ₀=3.

Fig. 20. Calculated solution of the 2-phase Whitham equations via characteristics for a DSW collision with large initial velocity. As time increases, the Riemann invariants r₃ and r₄ get closer to one another. This corresponds to the closure of one of the gaps in the spectrum of (10), and thus a degeneracy.

Fig. 21. Initial data for the case of a merger of two DSWs.

Fig. 22. Initial data regularization for the case of a merger of two DSWs.