Corrigan and Zambon (2023) showed that the good Boussinesq equation, a completely integrable nonlinear wave equation second order in time, admits momentum-conserving integrable defects: two fields on either side of a defect point, related by explicit nonlinear sewing conditions, with total energy and momentum conserved through a defect contribution. Unlike every previously analysed integrable defect model (sine-Gordon, KdV, nonlinear Schrödinger, affine Toda), where defects are purely transmitting, the Boussinesq defect can reflect solitons, capture them, and even decay by emitting one or two solitons. What the 2023 analysis did not establish is whether the model with the defect is completely integrable, in the sense proven for those earlier models (a Lax representation or an infinite hierarchy of independent conserved charges in Poisson involution surviving the defect). The problem is to settle this question for one and several defects — construct the integrable structure, or show that the Boussinesq defect model is the first counterexample among momentum-conserving defects — and thereby explain why this model alone permits defect reflection and decay.

Background

The Boussinesq equation was introduced in 1872 to describe long waves in shallow water. In the form used here — the 'good' Boussinesq equation, with positive dispersion coefficient — one introduces a potential field u(x,t)u(x,t) whose spatial derivative w=uxw=u_x satisfies

1c2 wtt−wxx−λ (w2)xx+κ2 wxxxx=0,κ2>0,\frac{1}{c^{2}}\,w_{tt}-w_{xx}-\lambda\,(w^{2})_{xx}+\kappa^{2}\,w_{xxxx}=0,\qquad\kappa^{2}>0,

where cc is a wave speed, κ\kappa a length scale and λ\lambda a coupling constant; equivalently uu satisfies 1c2utt−uxx−λ(ux2)x+κ2uxxxx=0\frac{1}{c^{2}}u_{tt}-u_{xx}-\lambda(u_x^{2})_x+\kappa^{2}u_{xxxx}=0. It is a rare example of a nonlinear wave equation that is second order in time yet completely integrable: it possesses a Lax representation (an auxiliary linear problem whose compatibility condition encodes the equation) and an infinite hierarchy of conserved quantities, supports soliton solutions such as u(x,t)=−3κγ(ν)λtanh⁡(γ(ν)2κ(x−νt))u(x,t)=-\frac{3\kappa\gamma(\nu)}{\lambda}\tanh\big(\frac{\gamma(\nu)}{2\kappa}(x-\nu t)\big) with γ(ν)=1−ν2/c2\gamma(\nu)=\sqrt{1-\nu^{2}/c^{2}} and ∣ν∣<c|\nu|<c, and, unusually, 'merging' or 'splitting' soliton solutions in which the number of solitons changes in the course of evolution (Rasin and Schiff 2017).

An integrable defect (also called a jump defect) is a point x=x0x=x_0 at which an integrable field theory is cut in two: one field uu lives on x<x0x<x_0 and another field vv on x>x0x>x_0, both satisfying the same bulk equations, with the fields and finitely many of their derivatives related at x0x_0 by sewing conditions. The Lagrangian framework of Bowcock, Corrigan and Zambon adds a defect Lagrangian D\mathcal{D} concentrated at x0x_0 and insists that the total energy and momentum — the bulk contributions from both half-lines plus defect contributions — be conserved even though placing the defect at a fixed point manifestly breaks translation invariance. In every model analysed before 2023 (sine-Gordon, KdV, nonlinear Schrödinger, affine Toda), imposing this yields sewing conditions that amount to Bäcklund transformations (relations mapping solutions of a nonlinear equation to other solutions) frozen at the defect point, and subsequent analyses showed that such momentum-conserving defects do not destroy complete integrability. In all of these models the defect is purely transmitting: an incoming soliton is transmitted with a delay, converted to its antisoliton, or captured — but never reflected, and a defect never decays into solitons.

Corrigan and Zambon (2023) carried out this programme for the good Boussinesq equation. The defect contribution has the form D=utF+vtG+H\mathcal{D}=u_t F+v_t G+H with F=αux+βvx+fF=\alpha u_x+\beta v_x+f, G=βux+αvx+gG=\beta u_x+\alpha v_x+g, α2=κ2/3c2\alpha^{2}=\kappa^{2}/3c^{2}, β2=4κ2/3c2\beta^{2}=4\kappa^{2}/3c^{2}, H=σ2(βux2+2αuxvx+βvx2)+ρ(ux+vx)+τH=\frac{\sigma}{2}(\beta u_x^{2}+2\alpha u_x v_x+\beta v_x^{2})+\rho(u_x+v_x)+\tau, where σ=σ0+μ(u−v)\sigma=\sigma_0+\mu(u-v) with μ=2λ/3β\mu=2\lambda/3\beta and ρ,τ\rho,\tau are fixed functions of σ\sigma; the two essential free parameters are σ0\sigma_0 and ρ0\rho_0, while f,gf,g are subject only to ∂f/∂v−∂g/∂u=σ/c2\partial f/\partial v-\partial g/\partial u=\sigma/c^{2} (a mild non-uniqueness). Total energy conservation then holds whatever the defect's details (the defect merely adds to the conserved energy), but requiring a suitably modified total momentum to be conserved as well fixes the form of D\mathcal{D}, and varying the action yields four sewing conditions at x=x0x=x_0 relating uxx,vxx,uxxx,vxxxu_{xx},v_{xx},u_{xxx},v_{xxx} to time derivatives and lower derivatives of both fields, the first two of which read

κ2uxx=αut+βvt+σ(βux+αvx)+ρ,κ2vxx=−(βut+αvt+σ(αux+βvx)+ρ),\kappa^{2}u_{xx}=\alpha u_t+\beta v_t+\sigma(\beta u_x+\alpha v_x)+\rho,\qquad \kappa^{2}v_{xx}=-\big(\beta u_t+\alpha v_t+\sigma(\alpha u_x+\beta v_x)+\rho\big),

(the full set is collected in Section 3.4 of that paper). With these conditions the authors verified total energy and momentum conservation and analysed the scattering. A single soliton is transmitted with a delay, or captured. Strikingly, however, the merging/splitting solutions display effects absent from every other integrable defect model: reflection from the defect, a defect holding a store of momentum and energy that is partially released by the emission of one or two solitons, and parameter ranges in which a soliton disappears altogether. Such behaviour is typical of a linear wave equation with a delta-function point impurity, not of integrable defects.

What the 2023 analysis established is thus a necessary condition (conservation of energy and momentum), not full integrability. Whether the Boussinesq model with such defects is completely integrable — whether an auxiliary linear problem (Lax pair) exists whose data incorporate the sewing conditions, or an infinite tower of independent conserved quantities in Poisson involution (i.e. pairwise commuting with respect to the Hamiltonian bracket) survives the defect, for one and also several defects — remains open. The question is further motivated by context: a perturbed Boussinesq equation models pulses in nerve membranes in the soliton theory of the nerve impulse (Heimburg and Jackson), which made the defect a candidate toy model of a synapse; and the quantum Boussinesq model is related to W3W_3 conformal field theory (whose symmetry algebra extends the Virasoro algebra of two-dimensional conformal field theory by an additional current). Those quantum and nerve-fibre questions are separate open problems noted below; the present problem is the classical integrability question posed by the source for one or more defects, where the authors note that a counterexample to the expected survival of integrability would itself be a worthwhile result.

Problem Statement

Determine whether the good Boussinesq equation with the Corrigan-Zambon momentum-conserving defect is completely integrable, and account within the resulting structure for the defect-scattering phenomena unique to this model.

Specifically, for the defect model defined by the bulk field equations on either side of x=x0x=x_0 and the sewing conditions of Corrigan and Zambon (2023), Section 3.4 (two of which are quoted in the background):

  1. Either construct a structure establishing complete integrability — for example, a Lax pair (zero-curvature representation) whose auxiliary linear problem reproduces the sewing conditions as matching conditions at x0x_0 with time-independent monodromy (the ordered product of the fundamental-solution matrices of the auxiliary problem across the two half-lines and the defect, whose time-independence generates the conserved quantities), a Bäcklund or dressing formulation incorporating the defect, an inverse-scattering solution of the initial-value problem with defect, or an explicit infinite hierarchy of independent conserved quantities in Poisson involution whose conservation follows from the bulk equations together with the sewing conditions — valid for one and also for several defects;

  2. Or demonstrate that the model fails such a criterion, which would constitute the first known counterexample among momentum-conserving defects of integrable models;

  3. In either case, explain the origin of the phenomena that motivated the question: why the Boussinesq defect can reflect, capture, and decay by emitting one or two solitons while all previously known integrable defects are purely transmitting.

The common objective is the complete classical integrable structure of the defect model; deliverables (1)-(2) are the alternative decidable core, and deliverable (3) is the interpretive component that a complete answer should supply through that structure. The source poses these within a list of six open problems; its further items (the quantum Boussinesq defect theory and the perturbed nerve-fibre model) are separate open problems, not part of this problem's completion boundary.

References

  1. E. Corrigan and C. Zambon, 'Adding integrable defects to the Boussinesq equation', J. Phys. A: Math. Theor. 56 (2023) 385701. DOI: 10.1088/1751-8121/aceec9; arXiv:2306.03772. https://arxiv.org/abs/2306.03772
  2. A. G. Rasin and J. Schiff, 'Bäcklund transformations for the Boussinesq equation and merging solitons', J. Phys. A: Math. Theor. 50 (2017) 325202. DOI: 10.1088/1751-8121/aa7af7; arXiv:1701.00243. https://arxiv.org/abs/1701.00243
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