In this work, we construct commutative rings of two variable matrix differential operators that are isomorphic to a ring of meromorphic functions on a rational manifold obtained from the CP 1×CP 1 by identification of two lines with the pole on a certain rational curve. The commutation condition for differential operators is equivalent to a system of non-linear equations in the operators’ coeff...

We consider the existence and propagation of nondi ractive and nondispersive spatiotemporal optical wavepackets in nonlinear Kerr media. We report analytically and con rm numerically the properties of spatiotemporal dark line solitary wave solutions of the (2+1)D nonlinear Schrödinger equation (NLSE). Dark lines represent holes of light on a continuous wave background. Moreover, we consider non...

In nonlinear science, it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and Painlevé waves. Actually, even if for the most well-known prototypical models such as the Kortewet-de Vries (KdV) equation and the Kadomtsev-Petviashvili (KP) equation, this kind of problem has not yet been solved. In this paper, the explic...

Based on the symbolic computation approach, multiple rogue wave solutions of generalized (2+1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili equation are studied. As an example, we present 1-rogue solutions, 3-rogue and 6-rogue solutions. Furthermore, some dynamics features obtained shown by 3D, contour density graphics.

We consider the fifth order Kadomtsev-Petviashvili I (KP-I) equation as ∂tu + α∂ 3 xu + ∂ 5 xu + ∂ −1 x ∂ 2 yu + uux = 0, while α ∈ R. We introduce an interpolated energy space Es to consider the well-posedeness of the initial value problem (IVP) of the fifth order KP-I equation. We obtain the local well-posedness of IVP of the fifth order KP-I equation in Es for 0 < s ≤ 1. To obtain the local ...

In this paper, a variable-coefficient symbolic computation approach is proposed to solve the multiple rogue wave solutions of nonlinear equation with variable coefficients. As an application, (2+1)-dimensional Kadomtsev-Petviashvili investigated. The are obtained and their dynamics features shown in some 3D contour plots.

A linearized implicit local energy-preserving (LEP) scheme is proposed for the KPI equation by discretizing its multi-symplectic Hamiltonian form with Kahan's method in time and symplectic Euler-box rule space. It can be implemented easily, also it less storage-consuming more efficient than fully methods. Several numerical experiments, including simulations of evolution line-soliton lump-type s...