Nonlocal symmetry, Darboux transformation and soliton–cnoidal wave interaction solution for the shallow water wave equation

Abstract

In classical shallow water wave (SWW) theory, there exist two integrable one-dimensional SWW equation [HirotaSatsuma (HS) type and Ablowitz-Kaup-Newell-Segur (AKNS) type] in the Boussinesq approximation. In this paper, we mainly focus on the integrable SWW equation of AKNS type. The nonlocal symmetry in form of square spectral function is derived starting from its Lax pair. Infinitely many nonlocal symmetries are presented by introducing the arbitrary spectrum parameter. These nonlocal symmetries can be localized and the SWW equation is extended to enlarged system with auxiliary dependent variables. Then Darboux transformation for the prolonged system is found by solving the initial value problem. Similarity reductions related to the nonlocal symmetry and explicit group invariant solutions are obtained. It is shown that the soliton-cnoidal wave interaction solution, which represents soliton lying on a cnoidal periodic wave background, can be obtained analytically. Interesting characteristics of the interaction solution between soliton and cnoidal periodic wave are displayed graphically.

Soliton, breather, lump and their interaction solutions of the ($2+1$)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

“…A set of systematic methods have been used in the literature to obtain reliable treatments of nonlinear evolution equations. So far, researchers have established several methods to find the exact solutions, including the inverse scattering transform [1], the Bäcklund transformation [2][3][4][5], the Darboux transformation [6][7][8][9][10][11][12][13][14], the Riemann-Hilbert approach [15][16][17] and Hirota's bilinear method [18][19][20][21][22][23][24][25][26][27][28], Jacobian elliptic function method and modified tanh-function method [29][30][31][32][33]. Each of these approaches has its features, Hirota's bilinear method is widely popular due to its simplicity and directness.…”

N-Soliton Solutions for the NLS-Like Equation and Perturbation Theory Based on the Riemann–Hilbert Problem

“…Thus, this method has been constantly developed [9][10][11][12][13][14][15]. Furthermore, there are many techniques and transformations for finding exact solutions for soliton equations, such as the Darboux transformation method [16][17][18], the Bäcklund transformation method [19,20], the Hirota bilinear method [21][22][23], the homogeneous balance method [24,25], Frobenius integrable decompositions [26][27][28], and Wronskian technology [29,30]. These methods have greatly promoted the development of soliton theory.…”

Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems

“…Regularisation is a commonly used method to solve ill-conditioned problems that causes regression coefficients to have smaller variance values, thus solving potentially ill-posed problems [21][22][23][24][25][26][27][28]. e Tikhonov regularisation (TR) method [21][22][23], truncated singular value method [24,25], kernel function-based regularisation method [26,27], and l 1 norm regularisation method [28] are often used to solve ill-posed problems. When estimating nonlinear parameters, iterative search methods such as the Gauss-Newton method, steepest gradient method, and LM method are commonly used [29][30][31][32][33][34].…”