Generalized Darboux transformation and rational soliton solutions for Chen–Lee–Liu equation

PINN deep learning for the Chen-Lee-Liu equation: Rogue wave on the periodic background

“…T with T being the matrix transpose are the distinct solutions of Lax Pair (3.1) related to λ k , and the seed solution is q [0] = Ae iθ , θ = ax − (aA 2 + a 2 )t, of which a and A being the complex parameters, then the N -th order analytic solutions for CLL equation (1.1) are written into the following determinant expression [9,10]…”

“…For CLL equation (1.1), as presented in Ref. [9,10], the rational solution, breather wave and rogue wave on the constant background have been constructed via expression (3.3) when N = 2n. However, we hereby try to construct the periodic wave and rogue periodic wave for CLL equation by taking N = 2n + 1 in expression (3.3).…”

“…Using the Hirota method, the exact N -soliton solution of the CLL equation was constructed [7,8]. The breather solution, rogue wave solution and rational soliton solution have been obtained based on the Darboux transformation(DT) [9,10]. The initial-boundary value problem for the CLL equation was analysed on the half line via the Fokas unified method [11].…”

Traveling Wave Solutions and Chaotic Motions for a Perturbed Nonlinear Schrödinger Equation with Power-Law Nonlinearity and Higher-Order Dispersions

“…Although these equations explain the pulse dynamics in optical fibers [19][20][21][22], some of these nonlinear models are non-integrable. In this context, various computational and analytical methods have been proposed and used in the past few decades, to examine many classes of Schrödinger equation [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. Nonetheless, these investigations reveal that the dynamic of solutions in non-integrable systems can be important and more complex.…”

The Rational Solutions and the Interactions of the N-Soliton Solutions for Boiti-Leon-Manna-Pempinelli-Like Equation

“…But the diverse nonlinear equations of mathematical physics was descried with Hirota bilinear equations [6] [7] [8] and generalized bilinear equations [9] [11] [12] [13], such as the KdV equations [10] [11] [14] [15], the BLMP equations [16] [17] [18], the NLS equation, the Boussinesq equation [19], the KP equations [9] [20] [21] and so on. There has been a growing attention on rational solutions [2] [3] [9] [10] [11] [22] to nonlinear differential equations in recent years. One kind of particular rational solutions are rogue wave solutions [24] [25] [26], which describe significant nonlinear wave phenomena in oceanography.…”