In this paper, the dimension of invariant subspaces admitted by nonlinear systems is estimated under certain conditions. It is shown that if the two-component nonlinear vector differential operator \mathbb{F} = (F^1 ,F^2 ) with orders {k 1, k 2} (k 1 ≥ k 2) preserves the invariant subspace W_{n_1 }^1 \times W_{n_2 }^2 (n_1 \geqslant n_2 ), then n 1 − n 2 ≤ k 2, n 1 ≤ 2(k 1 + k 2) + 1, where W_{n_q }^q is the space generated by solutions of a linear ordinary differential equation of order n q (q = 1, 2). Several examples including the (1+1)-dimensional diffusion system and Itô’s type, Drinfel’d-Sokolov-Wilson’s type and Whitham-Broer-Kaup’s type equations are presented to illustrate the result. Furthermore, the estimate of dimension for m-component nonlinear systems is also given.
The exact explicit traveling solutions to the two completely integrable sixth-order nonlinear equations KdV6 are given by using the method of dynamical systems and Cosgrove’s work. It is proved that these traveling wave solutions correspond to some orbits in the 4-dimensional phase space of two 4-dimensional dynamical systems. These orbits lie in the intersection of two level sets defined by two first integrals.
It is well-known that every member of the KdV hierarchy of equations can be obtained from the AKNS hierarchy of equations by imposing a simple reduction. The author finds that the reduction conditions of the potentials in the spectral problem can be replaced by adding additional eigenfunction equations to the spectral problem, and then shows that the restricted KdV flows, such as the Neumann system, the Garnier system and the generalized multicomponent Hénon-Hieles system, are a kind of special reductions of the restricted AKNS flows.
A class of non-semisimple matrix loop algebras consisting of triangular block matrices is introduced and used to generate bi-integrable couplings of soliton equations from zero curvature equations. The variational identities under non-degenerate, symmetric and ad-invariant bilinear forms are used to furnish Hamiltonian structures of the resulting bi-integrable couplings. A special case of the suggested loop algebras yields nonlinear bi-integrable Hamiltonian couplings for the AKNS soliton hierarchy.
The authors study the multi-soliton, multi-cuspon solutions to the Camassa-Holm equation and their interaction. According to the solution formula due to Li in 2004 and 2005, the authors give the proper choice of parameters for multi-soliton and multicuspon solutions, especially for n ≥ 3 case. The numerical method (the so-called local discontinuous Galerkin (LDG) method) is also used to simulate the solutions and give the comparison of exact solutions and numerical solutions. The numerical results for the two-soliton and one-cuspon, one-soliton and two-cuspon, three-soliton, three-cuspon, three-soliton and one-cuspon, two-soliton and two-cuspon, one-soliton and three-cuspon, four-soliton and four-cuspon are investigated by the numerical method for the first time, respectively.
All the possible equivalent barotropic (EB) laminar solutions are firstly explored, and all the possible non-EB elliptic circulations and hyperbolic laminar modes of rotating stratified fluids are discovered in this paper. The EB circulations (including the vortex streets and hurricane like vortices) possess rich structures, because either the arbitrary solutions of arbitrary nonlinear Poisson equations can be used or an arbitrary two-dimensional stream function is revealed which may be broadly applied in atmospheric and oceanic dynamics, plasma physics, astrophysics and so on. The discovery of the non-EB modes disproves a known conjecture.
The authors generalize the Cauchy matrix approach to get exact solutions to the lattice Boussinesq-type equations: lattice Boussinesq equation, lattice modified Boussinesq equation and lattice Schwarzian Boussinesq equation. Some kinds of solutions including soliton solutions, Jordan block solutions and mixed solutions are obtained.
By applying the fermionization approach, the inverse version of the bosonization approach, to the Sharma-Tasso-Olver (STO) equation, three simple supersymmetric extensions of the STO equation are obtained from the Painlevé analysis. Furthermore, some types of special exact solutions to the supersymmetric extensions are obtained.
This paper deals with the problem of the bounded traveling wave solutions’ shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equation for short. The authors employ the theory and method of planar dynamical systems to make comprehensive qualitative analyses to the above equation satisfied by the horizontal velocity component u(ξ) in the traveling wave solution (u(ξ), H(ξ)), and then give its global phase portraits. The authors obtain the existent conditions and the number of the solutions by using the relations between the components u(ξ) and H(ξ) in the solutions. The authors study the dissipation effect on the solutions, find out a critical value r*, and prove that the traveling wave solution (u(ξ),H(ξ)) appears as a kink profile solitary wave if the dissipation effect is greater, i.e., |r| ≥ r*, while it appears as a damped oscillatory wave if the dissipation effect is smaller, i.e., |r| < r*. Two solitary wave solutions to the WBK equation without dissipation effect is also obtained. Based on the above discussion and according to the evolution relations of orbits corresponding to the component u(ξ) in the global phase portraits, the authors obtain all approximate damped oscillatory solutions (ũ(ξ), \tilde H(ξ)) under various conditions by using the undetermined coefficients method. Finally, the error between the approximate damped oscillatory solution and the exact solution is an infinitesimal decreasing exponentially.
By means of the classical symmetry method, a hyperbolic Monge-Ampère equation is investigated. The symmetry group is studied and its corresponding group invariant solutions are constructed. Based on the associated vector of the obtained symmetry, the authors construct the group-invariant optimal system of the hyperbolic Monge-Ampère equation, from which two interesting classes of solutions to the hyperbolic Monge-Ampère equation are obtained successfully.