A previously derived equation and boundary condition [the generalized Kadomtsev‐Petviashvili system] is used to describe the propagation of stable solitary waves in open seas and marine straits. The GKP system is transformed, under specific geophysical conditions, into a simpler system that allows exact soliton type solutions. The curved wave crests corresponding to these solutions are plotted for several choices of the depth function and side boundaries.

A general class of fourth order scalar partial differential equations, invariant under the same group of local point transformations as the KP equation, is constructed. Invariance under this Kac-Moody-Virasoro type group implies that the corresponding nonlinear evolution equations must have the form (ut+32uux)x+34σuyy+u32xxH(K2, …, K10)=0, where H is an arbitrary function of nine elementary invariants obtained in this article.

A considerable amount of information is currently available on the creation and propagation of large solitary waves in marine straits. In order to be able to analyze such data we develop a theoretical model, extending previous one‐dimensional models to the case of straits with varying width and depth, and nonvanishing vorticity. Starting from the Euler equations for a three‐dimensional homogeneous incompressible inviscid fluid, we derive, in the quasi‐one‐dimensional long‐wave and shallow‐water approximation, a generalized KadomtsevPetviashvili (GKP) equation, together with its appropriate boundary conditions. In general, the coefficients of this equation depend on the form of the bottom and on the vorticity; the sides of the straits figure only in the boundary conditions. Under certain restrictions on the vorticity and the geometry of the straits we reduce the GKP equation to one of several completely integrable partial differential equations, in order to study the evolution of solitons which originate in the straits.

The Kadomtsev–Petviashvili (KP) equation (ut+3uux/2+ 1/4 uxxx)x +3σuyy/4=0 allows an infinite-dimensional Lie group of symmetries, i.e., a group transforming solutions amongst each other. The Lie algebra of this symmetry group depends on three arbitrary functions of time ‘‘t’’ and is shown to be related to a subalgebra of the loop algebra A(1)4. Low-dimensional subalgebras of the symmetry algebra are identified, specifically all those of dimension n≤3, and also a physically important six-dimensional Lie algebra containing translations, dilations, Galilei transformations, and ‘‘quasirotations.’’ New solutions of the KP equation are obtained by symmetry reduction, using the one-dimensional subalgebras of the symmetry algebra. These solutions contain up to three arbitrary functions of t.

It is shown that the symmetry algebra of the Kadomtsev-Petviashvili equation can be related to an infinite-dimensional subalgebra of the loop algebra [$\mathrm{SL}(5, R)\ensuremath{\bigotimes}R(t, {t}^{\ensuremath{-}1})\ensuremath{\bigoplus}[R(t, {t}^{\ensuremath{-}1})\frac{d}{\mathrm{dt}}]$. The algebra is used to generate new classes of solutions of the Kadomtsev-Petviashvili equation, depending on several arbitrary functions.

A new class of classical field theories, in 1+1 dimensions, is introduced, of the form iγμΨ, μ−mΨ−Ψ̄γμ(g1+g2γ5) ×ΨγμΨ−Ψ̄(g3+g4γ5)ΨΨ=0. It is shown that these theories are relativistically invariant; they do not, however, preserve parity in general, and thus could be used to describe the dynamics of weak interaction processes. The prolongation structure method is used to investigate the existence of pseudopotentials. When the coupling constants g3 and g4 are zero, the corresponding theory is then characterized by an infinite family of conservation laws and is thus completely integrable. For this very case, the Bäcklund map (pseudopotential) furnishes the equivalent of a Lax pair of operators as well as a nontrivial Bäcklund transformation and solutions of soliton type.

It is shown how the reduction method applied to a 2×2 Zakharov-Shabat system with appropriate meromorphic structure leads to the Thirring model as integrability conditions. Reducing the generic soliton-generating multi-Bäcklund transformations, the general multi-soliton solutions are explicitly derived.