We show a direct connection between a cellular automaton and integrable nonlinear wave equations. We also present the N-soliton formula for the cellular automaton. Finally, we propose a general method for constructing such integrable cellular automata and their N-soliton solutions.

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A new soliton cellular automaton is proposed. It is defined by an array of an infinite number of boxes, a finite number of balls and a carrier of balls. Moreover, it reduces to a discrete equation obtained from discrete modified Korteweg-de Vries equation through a limit. Algebraic expression of soliton solutions is also proposed.

When continuous symmetry is spontaneously broken, there appear Nambu-Goldstone modes (NGMs) with linear or quadratic dispersion relation, which is called type-I or type-II, respectively. We propose a framework to count these modes including the coefficients of the dispersion relations by applying the standard Gross-Pitaevskii-Bogoliubov theory. Our method is mainly based on (i) zero-mode solutions of the Bogoliubov equation originated from spontaneous symmetry breaking and (ii) their generalized orthogonal relations, which naturally arise from well-known Bogoliubov transformations and are referred to as "σ-orthogonality" in this paper. Unlike previous works, our framework is applicable without any modification to the cases where there are additional zero modes, which do not have a symmetry origin, such as quasi-NGMs, and/or where spacetime symmetry is spontaneously broken in the presence of a topological soliton or a vortex. As a by-product of the formulation, we also give a compact summary for mathematics of bosonic Bogoliubov equations and Bogoliubov transformations, which becomes a foundation for any problem of Bogoliubov quasiparticles. The general results are illustrated by various examples in spinor Bose-Einstein condensates (BECs). In particular, the result on the spin-3 BECs includes new findings such as a type-I-type-II transition and an increase of the type-II dispersion coefficient caused by the presence of a linearly-independent pair of zero modes.

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We exhaust all exact self-consistent solutions of complex-valued fermionic condensates in the 1+1 dimensional Bogoliubov-de Gennes and chiral Gross-Neveu systems under uniform boundary conditions. We obtain n complex (twisted) kinks, or grey solitons, with 2n parameters corresponding to their positions and phase shifts. Each soliton can be placed at an arbitrary position while the self-consistency requires its phase shift to be quantized by π/N for N flavors.PACS numbers: 11.10. Kk, 03.75.Ss, Introduction.-The search for inhomogeneous selfconsistent fermionic condensates including states such as the Fulde-Ferrell (FF) [1] and Larkin-Ovchinnikov (LO) [2] states having phase and amplitude modulations, respectively, in superconductors has attracted considerable attentions for more than half a century since theoretical predictions were made about their existence. While amplitude modulations are well studied in conducting polymers [3][4][5][6][7], the FFLO state is theoretically shown to be a ground state of superconductors under a magnetic field [8]. Recently, the FFLO state has also been discussed in the context of an ultracold atomic Fermi gas [9,10]. In general, inhomogeneous self-consistent fermionic condensates with a gap function and fermionic excitations can be treated simultaneously using the Bogoliubov-de Gennes (BdG) and gap equations [11]. The gap functions are real and complex for conducting polymers [12] and superconductors, respectively. In the quantum field theory, these systems correspond to the Gross-Neveu (GN) model [13] and the Nambu-Jona-Lasinio (or chiral GN) model [14], which were proposed as models of dynamical chiral symmetry breaking in 1+1 or 2+1 dimensions. Therefore, BdG and (chiral) GN systems have been studied and developed together from the viewpoint of both condensed matter physics and high energy physics (see Ref.[15] for a review). For instance, fermion number fractionization is one of the topics that has been studied from this viewpoint [16,17]. Recently, it has been shown that the solutions in 1+1 dimensions can be promoted to 3+1 dimensions [18,19], thereby leading to extensive study of the modulated phases of these systems in terms of quantum chromodynamics (QCD) [20].Inhomogeneous self-consistent solutions are often studied numerically because analytic solutions are generally difficult to obtain. However, several analytic solutions are available in the case of the real-valued condensates in 1+1 dimensions, which describe the conducting polymers and the real GN model. Under uniform boundary conditions at spatial infinities, a real kink was constructed by Dashen et. al. [21] by using the inverse scattering method, and later, it was reconstructed in polyacetylene [22] in the continuum limit of the