Symmetry and general symmetry groups of the coupled Kadomtsev–Petviashvili equation

Abstract

In this paper, the Lie symmetry algebra of the coupled Kadomtsev-Petviashvili (cKP) equation is obtained by the classical Lie group method and this algebra is shown to have a Kac-Moody-Virasoro loop algebra structure. Then the general symmetry groups of the cKP equation is also obtained by the symmetry group direct method which is proposed by Lou et al。From the general symmetry groups, the Lie symmetry group can be recovered and a group of discrete transformations can be derived simultaneously. Lastly, from a known simple solution of the cKP equation, we can easily obtain two new solutions by the general symmetry groups.

Modified (2+1)-dimensional displacement shallow water wave system and its approximate similarity solutions

“…To study the Lie point symmetry reductions of Eqs. (12), we can search for the transformations in the vector form of…”

“…For Eqs. (12) with k = 0, 1, 2, the Lie point symmetry components σ i (i = 1, 2, 3, 4, 5, 6) are in the forms of…”

“…( 2) can move in both directions and the velocity of the solitary wave is slightly less than that of the KdV soliton. Similar to the (1+1)-dimensional case, some (2+1)-dimensional models such as the Kadomtsev-Petviashvili (KP) equation [11,12] (u τ + 6uu ξ + u ξξξ ) ξ + u ηη = 0, (3) are used to describe the (2+1)-dimensional shallow water waves. For the KP equation, there is another new problem besides the previous two (moving in a single direction and having the larger velocity than the real shallow water waves), i.e., x and y are nonsymmetric in the model equation.…”

Research Progress on Nutrient Composition of Green Pepper

“…Green pepper (ZanthoxylumschinifoliumSiebetZucc) for the Rutaceae pepper plant, Also known as pepper, cliff pepper, wild pepper, green pepper, dog pepper, Because its fruit ripens for the cyan [1], mainly produced in Chongqing Jiangjin, Sichuan Jinyang, Emei, Liaoning Dandong, Yunnan Province on both sides of the Jinsha River and other places [2]. Green pepper peel is the raw material of essence and spice, seed is good woody oil, oil cake can be used as fertilizer or feed, leaf can be used as seasoning, edible or making pretzel tea instead of fruit, and it is also an important soil and water conservation tree in arid and semi-arid mountainous area [3].…”

Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrödinger equation with variable coefficients

“…In the past several years, in order to obtain exact solutions of NLS equations, many effective methods have been developed, such as the inverse scattering method, [19] the Bäcklund transformation, the Darboux transformation, [20] the Hirota method, [21] the Lie symmetry method, [22,23] the Adomian decomposition method, [24] various tanh methods, [25][26][27][28] and the generalized sub-equation expansion method. [29,30] Ever since Bluman and Cole [31] and later Olver and Rosenau [32] generalized the Lie approach to encompass symmetry transformations, lots of research referring to symmetry has been done, [33][34][35][36][37][38][39] and the study of symmetries, symmetry groups, symmetry reductions, and group invariant solutions of NLS equations has become one of the most exciting and extremely active areas of research. For example, Gagnon et al [40] studied the symmetries of a (1+1)-dimensional variable-coefficient NLS equation which involved three arbitrary complex functions of spacetime and reported that the dimension of the Lie point symmetry group G of the (1+1)-dimensional variable-coefficient NLS equation was 1 <dimG ≤ 5; Mansfield et al [41] applied a certain nonlinear generalization of Lie's linear method for finding infinitesimal symmetries to a generalization of the (3+1)-dimensional coupled NLS system; Li et al presented finite symmetry transformation groups of a (2+1)-dimensional cubic NLS equation and its corresponding cylindrical NLS equations in Ref.…”