Interacting Solitons, Periodic Waves and Breather for Modified Korteweg–de Vries Equation

Abstract

We theoretically demonstrate a rich and significant new families of exact spatially localized and periodic wave solutions for the modified Korteweg-de Vries equation. The model applies for the description of different nonlinear structures which include breathers, interacting solitons and interacting periodic wave solutions. A joint parameter which can take both positive and negative values of unity appeared in the functional forms of those closed form solutions, thus implying that every solution is determined for each value of this parameter. The results indicate that the existence of newly derived structures depend on whether the type of nonlinearity of the medium should be considered self-focusing or defocusing. The obtained nonlinear waveforms show interesting properties that may find practical applications.

New Painlevé Integrable (3+1)-Dimensional Combined pKP-BKP Equation: Lump and Multiple Soliton Solutions

“…However, a soliton solution is an analytic solution that is exponentially localized in all directions in space of 𝑥, 𝑦 and 𝑧 and time 𝑡. [6,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] Lump solutions arise when surface tension dominates the shallow water surface, as in plasmas, optical media and other physical applications. The basis of symbolic computation method, the generalized positive quadratic function, is a powerful technique to study lump solutions.…”

“…where 𝑎11 > 0 and, for simplicity, we select 𝛼 = 𝛽 = 𝛾 = 𝑎 = 𝑏 = 𝑐 = 1. The obtained parameters (27) generate the class of positive quadratic function solutions upon substituting ( 27) into (26). This, in turn, gives a first class of lump solutions to the (3+1)-dimensional pKP-BKP equation by using 𝑢 = 2(ln 𝑓 (𝑥, 𝑦, 𝑧, 𝑡))𝑥 as follows.…”

“…to guarantee a well-defined function 𝑓 (𝑥, 𝑦, 𝑧, 𝑡), its positiveness and the localization of 𝑢(𝑥, 𝑦, 𝑧, 𝑡) in all directions in the space, respectively. The obtained parameters (31) generate the class of positive quadratic function solutions upon substituting (31) in (26). This in turn will give a first class of lump solutions to the (3+1)-dimensional pKP-BKP equation by using 𝑢 = 2(ln 𝑓 (𝑥, 𝑦, 𝑧, 𝑡))𝑥.…”

“…Many effective approaches [12][13][14][15][16][17][18][19][20][21][22][23][24] have been used to examine the complete integrability of nonlinear evolution equations, with elastic or nonelastic interactions, aiming to attain new results in scientific areas. Significant solutions in mathematical physics, such as breather, lump and rogue wave solutions, have been derived using a variety of efficient techniques, such as the algebraic-geometric method, the inverse scattering method, the Bäcklund transformation method, the Painlevé analysis, Lax integrability, the Darboux transformation method, [15][16][17][18][19][20][21][22] the Hirota bilinear method, [1][2][3][4][5][6][22][23][24][25][26][27][28][29][30][31][32][33][34][35] and other powerful schemes. In Ref.…”

Three-Soliton Interactions and the Implementation of Their All-Optical Switching Function

“…On the other hand, in optical communication systems, in order to improve the bit rate, each optical soliton is input into the fiber at smaller intervals. [18][19][20][21] However, this can cause interactions between adjacent optical soli-tons, which can cause distortion of some optical solitons, leading to a decrease in the bandwidth of the optical communication system and weakening its communication capability. [22][23][24][25] In order to improve the communication capability of the system, it is necessary to study the interaction between optical solitons in the system.…”