[Show abstract][Hide abstract] ABSTRACT: In this paper, a multiple vortex interaction model (MVIM) is proposed to describe the possible stream–vorticity interaction (SVI) and the vorticity–vorticity interaction (VVI) among vortices. The symmetries and conservation laws of the MVIM show that the SVI preserves the momenta, the vortex momenta and the energies of every vortex, and the interaction energies of every two vortices. However, the VVI destroys the energy conservation property for every vortex. Some special types of exact vortices and vortex source solutions including multiple point vortices, vortex dipoles, vortex multi-poles, fractal cyclons, fractal cyclon dipoles and Bessel vortices (BV) are presented. A special theoretical solution, the first BV is just the usual modon solution which can be used to describe the so-called atmospheric blocking. The second BV is supported by an atmospheric observation, saddle field, occurred over the North Pacific on 26 March, 2009. The characteristic features of vortex interactions are discussed under the MVIM without the ββ-effect via numerical simulations. Several interaction patterns such as merging, separation, mutual orbiting and absorption are reported. Those interaction behaviors are well consistent with some known fluid mechanical experiments and meteorologic observations.
Nonlinear Analysis Real World Applications 10/2012; 13(5):2079–2095. DOI:10.1016/j.nonrwa.2012.01.004 · 2.52 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We construct conservation laws of the equation family which possesses the same infinite dimensional Kac–Moody–Virasoro symmetry algebra as the Kadomtsev–Petviashvili (KP) equation. The conservation laws are calculated up to second-order group invariants and described by two arbitrary functions of six variables and one arbitrary function with four variables.
Physics Letters A 04/2010; 374(15):1704-1711. DOI:10.1016/j.physleta.2010.02.029 · 1.68 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Generally, natural scientific problems are so complicated that one has to establish some effective perturbation or nonperturbation theories with respect to some associated ideal models. We construct a new theory that combines perturbation and nonperturbation. An artificial nonlinear homotopy parameter plays the role of a perturbation parameter, while other artificial nonlinear parameters, which are independent of the original problems, introduced in the nonlinear homotopy models are nonperturbatively determined by means of the principle of minimal sensitivity. The method is demonstrated through several quantum anharmonic oscillators and a non-hermitian parity-time symmetric Hamiltonian system. In fact, the framework of the theory is rather general and can be applied to a broad range of natural phenomena. Possible applications to condensed matter physics, matter wave systems, and nonlinear optics are briefly discussed.
Chinese Physics Letters 11/2009; 26(11). DOI:10.1088/0256-307X/26/11/110201 · 0.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,
which arises from fluid dynamics. We summarize the general formulas for similarity reduction solutions and similarity reduction
equations of different orders, educing the related homotopy series solutions. Zero-order similarity reduction equations are
equivalent to the Painlevé IV type equation or Weierstrass elliptic equation. Higher order similarity solutions can be obtained
by solving linear variable coefficients ordinary differential equations. The auxiliary parameter has an effect on the convergence
of homotopy series solutions. Series solutions and similarity reduction equations from the approximate symmetry method can
be retrieved from the approximate homotopy symmetry method.
Science in China Series G Physics Mechanics and Astronomy 08/2009; 52(8):1169-1178. DOI:10.1007/s11433-009-0181-3 · 1.41 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A type of (2+1)-dimensional nonlinear Schrödinger equation with spatially inhomogeneous nonlinearity and an external potential is studied. It is found that special external potentials and spatially nonlinearities can support nonlinear localized waves.
Chinese Physics Letters 03/2009; 26(3). DOI:10.1088/0256-307X/26/3/030502 · 0.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A novel method based on nonlinear homotopy analysis is proposed to study physical problems with strong perturbations. To improve the validity of the method, the principle of minimal sensitivity (PMS) should be introduced. The procedure of the method is analytical and is systematically described through an example, the energy eigenvalue problem of anharmonic oscillators. Highly accurate numerical results show the validity of the nonsensitive nonlinear homotopy approach (NNHA) for different perturbation parameters from moderate to very strong. Additional parameters can be inserted in the method and the parameters should be fixed via PMS. It is quite interesting that the well-known LDE method is just a linear reduction of the NNHA. The method is easy to implement and can be extended to explore problems in other branches of physics and other scientific fields.
[Show abstract][Hide abstract] ABSTRACT: Generally, natural scientific problems are so complicated that one has to
establish some effective perturbation or nonperturbation theories with respect
to some associated ideal models. In this Letter, a new theory that combines
perturbation and nonperturbation is constructed. An artificial nonlinear
homotopy parameter plays the role of a perturbation parameter, while other
artificial nonlinear parameters, of which the original problems are
independent, introduced in the nonlinear homotopy models are nonperturbatively
determined by means of a principle minimal sensitivity. The method is
demonstrated through several quantum anharmonic oscillators and a non-hermitian
parity-time symmetric Hamiltonian system. In fact, the framework of the theory
is rather general that can be applied to a broad range of natural phenomena.
Possible applications to condensed matter physics, matter wave systems, and
nonlinear optics are briefly discussed.