Nonlocal KdV Equation: Form Local To Nonlocal

Nonlinear phenomena are ubiquitous in nature,for instance,the vortices of the atmosphere and strange water waves in the oceans,etc.The nonlinear Korteweg-de Vries(KdV)equation can describe rich nonlinear physical phenomena,so how to get the analytical solutions of the KdV equation and apply them to the actual physical phenomena is a very important work.In this thesis,we take the nonlinear KdV equation as the core and use the symbolic computation software Maple to study the exact solutions in local and nonlocal cases.The details are as follows:Chapter 1 introduces the research background,the present research status,and some basic knowledge.Finally,it briefly expounds the research content,innovation and organization structure of this thesis.Chapter 2 demonstrates that the finite transform group can be obtained by the localization of the infinite number of residues and the point Lie symmetry approach,which is equivalent to the second kind of the multiple Darboux-B¨acklund transformation.The nfold Darboux-B¨acklund transformation formula and some soliton solutions are obtained.Chapter 3 first derives the nonlocal variable coefficient KdV equation with shifted parity and delayed time reversal from the two vortex system in the atmosphere or ocean system.Then a none auto-B¨acklund transform is constructed to convert the nonlocal variable coefficient KdV equation into its corresponding nonlocal constant coefficient KdV equation.By solving the non-local constant coefficient KdV equation,the solitary wave solutions and the periodic wave solutions of the original equation are obtained.Finally,an approximate analytical solution of the two dipole blocking events with a life cycle in the atmosphere is given theoretically.Chapter 4 studies the exact solutions of the nonlocal complex modified KdV(mKdV)equation.The bilinear form of the nonlocal complex mKdV equation is obtained by using the bilinear method.By exploiting the hypothesis of the expansion function,we obtain its breathers and rogue waves.The elliptic function solutions are obtained from the tanh function expansion method.Chapter 5 gives the bilinear structures of some common local and nonlocal equations.On the Maple platform,the general procedures for solving the breathers and rogue waves of the nonlocal complex mKdV equation by the bilinear method is realized.Chapter 6 summarizes the full text and looks forward to the future research work.