On Bilinear Narrow Operators

In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator T:E×F→W defined on the Cartesian product of vector lattices E and F and taking values in a vector lattice W is narrow if the partial operators Tx and Ty are narrow for all x∈E,y∈F. We prove that, for order-continuous Köthe–Banach spaces E and F and a Banach space X, the classes of narrow and weakly function narrow bilinear operators from E×F to X are coincident. Then, we prove that every order-to-norm continuous C-compact bilinear regular operator T is narrow. Finally, we show that a regular bilinear operator T from the Cartesian product E×F of vector lattices E and F with the principal projection property to an order continuous Banach lattice G is narrow if and only if |T| is.

Abundant interaction between lump and k-kink, periodic and other analytical solutions for the (3+1)-D Burger system by bilinear analysis

In this paper, we study the (3+1)-dimensional Burger system which is considered in soliton theory and generated by considering the Hirota bilinear operators. The bilinear frame to the Burger system by using the multi-dimensional Bell polynomials is constructed. Also, based on the binary Bäcklund transformations, the generalized Bell polynomials are written. We retrieve some novel exact analytical solutions, containing interaction between lump and two kink wave solutions, interaction between lump and periodic wave solutions, interaction between stripe and periodic solutions, breather wave solutions, cross-kink wave solutions, interaction between kink and periodic wave solutions, multi-wave solutions, and finally solitary wave solutions for the (3+1)-dimensional Burger system by Maple symbolic computations. The required conditions of the analyticity and positivity of the solutions can be easily achieved by taking special choices of the involved parameters. The main ingredients for this scheme are to recover the Hirota trilinear forms and their generalized equivalences.