Ordering-independent period matrix of the Case (b) separable N=2 sine-Gordon breather

Two kinds of separable two-phase (N=2) solutions of the classical sine-Gordon equation, arising from the two inequivalent ways of connecting the same four spectral branch points into branch cuts, were shown by Kuo and Hu (arXiv:1210.2181) to be related by a Landen (degree-two modular) transformation of the genus-2 period matrix. That work nevertheless left the period matrix of the Case (b) breather ambiguous: substituting the Forest–McLaughlin loop relations into the period integrals in different orders yields either Re B_11 = 1/2 with Re B_11 ≠ Re B_21, or Re B_11 = Re B_21 — an ordering dependence attributed to the analytic continuation from kink to breather spectral data — and no first-principles rule was given for choosing between the two branch-cut structures. The problem is to determine the correct, ordering-independently derived genus-2 period matrix for the Case (b) separable N=2 breather, deciding whether the half-integer real part is a genuine consequence of the analytic continuation or an artifact of the approximate loop relations. No prior work is known to address this; the problem is treated as open.

Background

The classical sine-Gordon equation (SGE), $\phi_{tt}-\phi_{xx}+\sin\phi=0$ in relativistic conventions, is the paradigmatic integrable nonlinear wave equation: among its soliton solutions are kinks (topological charge carriers), antikinks, and breathers (soliton–antisoliton bound states). Beyond finitely many solitons, the SGE admits quasiperiodic finite-gap (also called algebro-geometric or N-phase) solutions: for phase number N the solution is expressed through a Riemann theta function $\theta(\mathbf{z}\mid B)$ — a quasi-periodic function of the N phase variables built from the period matrix $B$ defined below — associated with a hyperelliptic Riemann surface: the spectral curve of the linear eigenvalue problem (the Lax pair) whose compatibility with its partner equation reproduces the SGE, of genus N (informally, a surface with N handles). The period matrix $B$ is obtained by integrating the N holomorphic differentials $\omega_i$ on the curve, normalized so that $\oint_{a_j}\omega_i=\delta_{ij}$ over the canonical a-cycles, along the complementary b-cycles: $B_{ij}=\oint_{b_j}\omega_i$. For a fixed curve with a fixed cycle basis, $B$ is mathematically unique, symmetric, and has positive-definite imaginary part; the theta representation of the solution depends on this normalization. Forest and McLaughlin's 'concrete viewpoint' (J. Math. Phys. 23, 1248 (1982)) made this spectral theory explicit for the periodic SGE, tabulating loop relations — approximate identities between the cycle integrals, expressed through complete elliptic integrals — that allow the entries of $B$ to be assembled from genus-1 data (their Appendix D, Table II). Real N-phase SGE solutions were subsequently classified by Ercolani and Forest (Commun. Math. Phys. 99, 1 (1985)), following the modulational-stability analysis of two-phase wavetrains by Ercolani, Forest, and McLaughlin (Stud. Appl. Math. 71, 97 (1984)); numerical theta solutions with period matrices were computed by Flesch, Forest, and Sinha (Physica D 48, 169 (1991)). Within this theory, Forest, Sheu, and Sinha (SIAM J. Appl. Math. 52, 746 (1992)) isolated the spatially periodic breather-train sector in which two distinct kinds of separable N=2 configurations arise. 'Separable' means the genus-2 problem decomposes into two independent genus-1 (N=1) problems, so that the theta function factorizes; the two kinds — Case (a) and Case (b) in the source — correspond to the two inequivalent ways of connecting the same four branch points into branch cuts: disjoint cuts (or cuts meeting on the negative real axis) versus one cut nested inside the other. Kuo and Hu (arXiv:1210.2181; N=1 predecessor arXiv:1208.2798) showed that the two kinds are related by a Landen transformation — the classical degree-2 transformation of elliptic moduli, realized here as an integer symplectic modular transformation of the genus-2 period matrix — and studied the kink and breather subsectors of each case. Breather spectral data is reached from kink data by analytic continuation of the branch-point parameters $\phi_i\to\pi+i\eta_i$ (with real $\eta_i$). Two points in that work are relevant here. First, when the period matrix of the Case (b) breather is computed by substituting the loop relations into the integral expressions for its entries, the result depends on the order of substitution: one ordering yields $\operatorname{Re}B_{11}=\tfrac12\neq\operatorname{Re}B_{21}=\operatorname{Re}B_{12}$, while for the kink of the same case $\operatorname{Re}B_{11}=\operatorname{Re}B_{21}$; the authors state this 'seem[s] contradiction', attribute it to the analytic continuation, and note that the analogous substitution for $B_{22}$ is ordering-independent — but they give no canonical prescription, and this ordering ambiguity remains unresolved. Second, the choice of branch cuts is, in their words, 'man-made' — a 'spontaneously symmetry [breaking]' step needed only to compute loop integrals — and no principle is given that would select one structure over the other, even though the constructed Landen transformation shows both are legitimate solution families.

Problem Statement

Determine the correct genus-2 period matrix of the Case (b) separable N=2 sine-Gordon breather — the spectral data obtained from the Case (b) kink configuration by the analytic continuation $\phi_i\to\pi+i\eta_i$ — and give an ordering-independent derivation of it. Concretely: (i) decide whether the real-part structure is $\operatorname{Re}B_{11}=\tfrac12\neq\operatorname{Re}B_{21}=\operatorname{Re}B_{12}$ or $\operatorname{Re}B_{11}=\operatorname{Re}B_{12}=\operatorname{Re}B_{21}$, i.e. whether the half-integer entry is a genuine consequence of the analytic continuation (for example a theta-characteristic shift, meaning a half-integer offset of the theta-function argument) or an artifact of substituting the approximate Forest–McLaughlin loop relations at inconsistent positions; and (ii) supply a canonical prescription — for instance, direct evaluation of the hyperelliptic period integrals on the spectral curve, or a definite substitution/regularization rule for the loop relations — that yields every entry of $B$ for the Case (b) breather unambiguously, removing the dependence on the order in which the loop relations are substituted. The objective is a consistent, canonical theta-functional description of the Case (b) separable N=2 breather, with a uniquely defined period matrix compatible with the modular/Landen framework relating the two branch-cut structures.

References

1. Nan-Hong Kuo, C. D. Hu, 'Modular and Landen transformation between two kinds of separable solutions of Sine Gordon equation in N=2', arXiv:1210.2181 (2012; v3 2013), https://arxiv.org/abs/1210.2181
2. Nan-Hong Kuo, C. D. Hu, 'Modular and Landen transformations between finite solutions of Sine-Gordon Equation with phase N=1', arXiv:1208.2798 (2012), https://arxiv.org/abs/1208.2798
3. Nan-Hong Kuo, C. D. Hu, 'Analytical methods of asymmetry double sine-Gordon equation in infinite one-dimensional system', arXiv:1210.4706 (2012), https://arxiv.org/abs/1210.4706
4. M. Gregory Forest, David W. McLaughlin, 'Spectral theory for the periodic sine-Gordon equation: A concrete viewpoint', Journal of Mathematical Physics 23, 1248–1277 (1982), doi:10.1063/1.525509, https://doi.org/10.1063/1.525509
5. Nicholas Ercolani, M. Gregory Forest, David W. McLaughlin, 'Modulational Stability of Two-Phase Sine-Gordon Wavetrains', Studies in Applied Mathematics 71, 97–101 (1984), doi:10.1002/sapm198471291, https://doi.org/10.1002/sapm198471291
6. Nicholas M. Ercolani, M. Gregory Forest, 'The geometry of real sine-Gordon wavetrains', Communications in Mathematical Physics 99, 1–49 (1985), doi:10.1007/BF01466592, https://doi.org/10.1007/BF01466592
7. Randy Flesch, M. Gregory Forest, Amarendra Sinha, 'Numerical inverse spectral transform for the periodic sine-gordon equation: Theta function solutions and their linearized stability', Physica D 48, 169–231 (1991), doi:10.1016/0167-2789(91)90058-H, https://doi.org/10.1016/0167-2789(91)90058-H
8. M. Gregory Forest, S.-P. Sheu, Amarendra Sinha, 'Frequency and Phase Locking of Spatially Periodic Perturbed Sine-Gordon Breather Trains', SIAM Journal on Applied Mathematics 52, 746–761 (1992), doi:10.1137/0152043, https://doi.org/10.1137/0152043
9. Fritz Gesztesy, Helge Holden, 'A combined sine-Gordon and modified Korteweg-de Vries hierarchy and its algebro-geometric solutions', arXiv:solv-int/9707010 (1997), https://arxiv.org/abs/solv-int/9707010