Multi-soliton solutions with nonorthogonal null vectors and integrability

Construct multi-soliton solutions of the deformed sine-Gordon theory in 1+1-dimensional anti-de Sitter space using hyperbolic plane waves $(X\cdot\xi_i)^{mR}$ associated with nonorthogonal null vectors, and determine whether the theory is integrable and in what sense.

Background

The paper studies a curvature-deformed sine-Gordon equation on maximally symmetric spaces. Previous work constructed single-soliton solutions in anti-de Sitter space using powers of hyperbolic plane waves associated with mutually orthogonal null vectors. The authors explicitly ask whether relaxing the orthogonality condition can produce multi-soliton configurations in 1+1 dimensions.

The same question is connected to the broader issue of whether the curved-space theory retains an integrable structure comparable to the ordinary flat-space sine-Gordon theory, where integrability is associated with infinitely many conserved quantities and multi-soliton solutions.

References

Is it possible to construct multi-soliton solutions in $1+1$ dimensions using hyperbolic plane waves $ (X \cdot \xi_i ){m R}$ with null vectors that are not orthogonal, $(\xi_i \cdot \xi_i)=0$ and $(\xi_i \cdot \xi_j) \ne 0$ for $i \neq j$? Is the theory integrable, and in what sense?

— One more Sine-Gordon soliton in AdS  (2608.19859 - Diakonov, 20 Aug 2026) in Section 1, Introduction

Is it possible to generalize the approach to studying soliton-like solutions to other nonlinear models?

— One more Sine-Gordon soliton in AdS  (2608.19859 - Diakonov, 20 Aug 2026) in Section 1, Introduction

We also note that this is still a single-soliton solution, and we have not yet been able to find multi-soliton solutions with either null or non-null vectors.

— One more Sine-Gordon soliton in AdS  (2608.19859 - Diakonov, 20 Aug 2026) in Section 3, New solution

However, we cannot exclude the possibility that multi-soliton solutions in hyperbolic spaces cannot exist in principle.

— One more Sine-Gordon soliton in AdS  (2608.19859 - Diakonov, 20 Aug 2026) in Section 4, Conclusion and acknowledgments