Exact analysis and elastic interaction of multi-soliton for a two-dimensional Gross-Pitaevskii equation in the Bose-Einstein condensation

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.…”

“…The basic approaches to soliton solutions include the inverse scattering transform, [1][2][3][4][5][6][7][8][9][10] the Riemann-Hilbert technique, [11][12][13][14][15][16][17][18][19][20] the Darboux transformation, [16][17][18][19][20][21][22][23][24][25] and the Hirota direct method. Significant solutions in mathematical physics, such as breather, complexion, lump and rogue wave solutions, are particular reductions of soliton solutions for different situations.…”

“…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.…”

Propagation characteristics of bright and mixed solitons based on the variable coefficient (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation

“…孤子这一概念在如非线性光学、原子物 理、凝聚态物理等诸多领域得到关注和应 用 [1][2][3][4] 。由于通常在介质中传播的光波或 脉冲其电磁包络非单色性，各部分会因为 传播速度不同而导致包络展宽。如在空间维 度，光波因为衍射效应使光束半径增大，在 时间维度则由于群速度色散致使脉冲展宽。 因而，在特殊条件下利用传播介质的非线性 效应或直接使用光纤放大器实现衰减补偿， 使非线性效应与衍射效应和色散效应达到平 衡，可以弥补线性传输过程中光波或脉冲的 发散 [5][6][7][8] 。这样的孤子具有稳定的、局域的、 类粒子的特性，被叫做光孤子。 根据光孤子所在的色散区域不同，光孤 子可分为明孤子和暗孤子。光孤子具有一系 列重要的应用价值 [9][10][11][12][13][14][15][16][17][18][19] ，例如将光孤子当作 比特信息应用于超快光学数字逻辑系统；还 可以研究超快光与物质之间的非线性相互作 用；应用于远程通信系统时，其中孤子的传 播可以用三次-五次复金兹堡-朗道(CGL)方 程来建模 [20][21][22] 。 三次-五次 CGL 方程是对被动锁模激光 器中区域动力学的一个持续估计 [23] CGL 方程 [24] ：…”

Effective Control of Three Soliton Interactions for the High-Order Nonlinear Schrödinger Equation

“…With the development of fiber optic communication, as a carrier for transmitting information, how to generate femtosecond optical solitons with richer frequency components and narrower pulses has become one of the main considerations for researchers. [1][2][3][4][5][6][7][8] Therefore, generation and transmission of femtosecond optical solitons in optical communication systems have important scientific significance, and the relevant results will provide a certain theoretical reference for further obtaining ultra-short and ultra-high energy optical solitons. [9][10][11][12][13][14] For optical communication systems, the nonlinear Schrödinger (NLS) equation describes the transmission process of picosecond solitons in single-mode fibers.…”