Lie symmetry, nonlocal symmetry analysis, and interaction of solutions of a (2+1)-dimensional KdV–mKdV equation

Nonlocal symmetries, nonlocally related systems, similarity solutions and conservation laws of the Tzitzéica-Dodd-Bullogh equation

“…It is easy to show that v2 is a local symmetry of TDB equation. To verify that v1 is a nonlocal symmetry, we write the system (17) as…”

New analytical solutions and integrability for the (2 + 1)-dimensional variable coefficients generalized Nizhnik-Novikov-Veselov system arising in the study of fluid dynamics via auto-Backlund transformation approach

“…To assure the analytic solvability of NLPDEs, integrability is a crucial aspect. In recent years, the various applications of integrability and localized wave solutions of numerous NLPDEs can be noticed such as, the rogue wave and multiple lump solutions in the form of Grammian formula for the (2 + 1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili (BKP) equation have been obtained by using polynomial function approach in [6]; multiple lump molecules and interaction solutions for the Kadomtsev-Petviashvili I equation are employed by utilizing non-homogeneous polynomial technique in [7]; lump chain solutions for the (2 + 1)dimensional BKP equation have been determined by using the τ-function in the form of Grammian formula in [8]; the soliton-cnoidal wave and lump-type solutions for (2 + 1)-dimensional KdV-mKdV equation are derived by utilizing Lie symmetry analysis and Bäcklund transformation approaches in [9]. The theoretical studies in nonlinear evolution equations have various applications in the diverse area of science and technology, such as: the electrohydrodynamics of a thin suspended liquid film model, which describes an incompressible fluid is examined by the perturbation technique in [10]; the granular model arising in the fluid dynamics has been solved by Painlevé analysis, Bäcklund transformation, Jacobi elliptic function, and tanh function methods in [11]; the magma equation arising in porous media is examined by the Cole-Hopf transformation method in [12]; the turbulent magnetohydrodynamic model in plasma turbulence has been solved using complex ansatz method in [13]; the compressible magnetohydrodynamic equations in cold plasma is examined by using the reductive perturbation method in [14].…”

Breathers, Transformation Mechanisms and Their Molecular State of a (3+1)-Dimensional Generalized Yu–Toda–Sasa–Fukuyama Equation

“…In the field of physics and mechanics, many scholars have become devoted to investigating the exact solutions of nonlinear integrable systems, which include solitons [1][2][3][4], breathers [5][6][7], lumps [8][9][10][11][12], quasi-periodic wave solutions [13][14][15][16], and so on. Scholars have undertaken a lot of work on exact solutions, and have summarized some effective methods, such as the Bäcklund transformation [17][18][19], inverse scattering method [20][21][22], Darboux transformation [23,24], algebraic geometry theory [25,26], Hirota's bilinear method [27,28] and Lie symmetry analysis [29][30][31][32].…”