Studies in Applied Mathematics

Liu, Hanze

doi: 10.1111/sapm.12011pmid: N/A

This paper is concerned with the fractional fifth‐order KdV types of equations, the complete group classification is performed on the general fractional fifth‐order partial differential equation (FPDE), which includes a lot of important fifth‐order fractional differential equations and nonlinear evolution equations (NLEEs) as its special cases. In particular, all of the point symmetries of the fifth‐order nonlinear evolution equation are presented with respect to the arbitrary parameters of the equation. In the sense of point symmetry, all of the vector fields of the equations are obtained. Then, the symmetry reductions are provided, and the exact analytic solutions to the general fifth‐order KdV equations are investigated.

Jiang, Yan; Tian, Bo; Li, Min; Wang, Pan

doi: 10.1111/sapm.12012pmid: N/A

In this paper, we will investigate a (2+1)‐dimensional breaking soliton (BS) equation for the (2+1)‐dimensional collision of a Riemann wave with a long wave in certain fluids. Using the Bell polynomials and an auxiliary function, we derive a new bilinear form for the (2+1)‐dimensional BS equation, which is different from those in the previous literatures. One‐, two‐ and N‐shock‐wave solutions are obtained with the Hirota method and symbolic computation. One shock wave is found to be able to stably propagate. Two shock waves are observed to have the parallel collision, oblique collision, and stable propagation of the V‐type structure. In addition, we present the collision between one shock wave and V‐type structure, and the collision between two V‐type structures.

Chesnokov, A. A.; Khe, A. K.

doi: 10.1111/sapm.12013pmid: N/A

Asymptotic analysis for small long‐wave perturbations of a given stationary shear flow of an ideal fluid with free boundary as t→∞ is performed. It is shown that small disturbances of the flow are attracted to periodic solution in the case where the governing equations are hyperbolic on the main shear flow solution. A class of shear flows for which Landau damping is realizable, is described. Analytical results obtained are validated by numerical calculations.

Duran, Angel; Dutykh, Denys; Mitsotakis, Dimitrios

doi: 10.1111/sapm.12015pmid: N/A

Surface water waves in ideal fluids have been typically modeled by asymptotic approximations of the full Euler equations. Some of these simplified models lose relevant properties of the full water wave problem. One of these properties is the Galilean symmetry, that is, the invariance under Galilean transformations. In this paper, a mechanism to incorporate Galilean invariance in classical water wave models is proposed. The technique is applied to the Benajmin–Bona–Mahony (BBM) equation and the Peregrine (classical Boussinesq) system, leading to the corresponding Galilean invariant versions of these models. Some properties of the new equations are presented, with special emphasis on the computation and interaction of solitary wave solutions. A comparison with the Euler equations demonstrates the relevance of the Galilean invariance in the description of water waves.

Chen, Guanwei; Ma, Shiwang

doi: 10.1111/sapm.12016pmid: N/A

We study the discrete nonlinear equation Lun−ωun=±χngn(un),n∈Z,where ω∉σ(L) (the spectrum of L) and gn(s) is asymptotically linear as |s|→∞ for all n∈Z. We obtain the existence of ground state solitons and the existence of infinitely many pairs of geometrically distinct solitons of this equation. Our method is based on the generalized Nehari manifold method developed recently by Szulkin and Weth. To the best of our knowledge, this technique has not been used for discrete equations with saturable nonlinearities.