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.

Asymptotic analysis for small long-wave perturbations of a given stationary shear flow of an ideal fluid with free boundary as 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.

In this paper, we discuss the conditions under which the coupled KdV and coupled Harry Dym hierarchies possess inverse (negative) parts. We further investigate the structure of nonlocal parts of tensor invariants of these hierarchies, in particular, the nonlocal terms of vector fields, conserved one-forms, recursion operators, Poisson and symplectic operators. We show that the invertible coupled KdV hierarchies possess Poisson structures that are at most weakly nonlocal while coupled Harry Dym hierarchies have Poisson structures with nonlocalities of the third order.

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.

The conditional Lie–Bäcklund symmetry method is used to study the invariant subspace of the nonlinear diffusion equations with convection and source terms. We obtain a complete list of canonical forms for such equations which admit higher order conditional Lie–Bäcklund symmetries and multidimensional invariant subspaces. The functionally generalized separable solutions to the resulting equations are constructed due to the corresponding symmetry reductions. For most of the cases, they are reduced to solving finite-dimensional dynamical systems.

The usual Cauchy matrix approach starts from a known plain wave factor vector and known dressed Cauchy matrix . In this paper, we start from a determining matrix equation set with undetermined and . From the determining equation set we can build shift relations for some defined scalar functions and then derive lattice equations. The determining equation set admits more choices for and and in the paper we give explicit formulae for all possible and . As applications, we get more solutions than usual multisoliton solutions for many lattice equations including the lattice potential KdV equation, the lattice potential modified KdV equation, the lattice Schwarzian KdV equation, NQC equation, and some lattice equations in ABS list.

As a first step toward a fully two-dimensional asymptotic theory for the bifurcation of solitons from infinitesimal continuous waves, an analytical theory is presented for line solitons, whose envelope varies only along one direction, in general two-dimensional periodic potentials. For this two-dimensional problem, it is no longer viable to rely on a certain recurrence relation for going beyond all orders of the usual multiscale perturbation expansion, a key step of the exponential asymptotics procedure previously used for solitons in one-dimensional problems. Instead, we propose a more direct treatment which not only overcomes the recurrence-relation limitation, but also simplifies the exponential asymptotics process. Using this modified technique, we show that line solitons with any rational line slopes bifurcate out from every Bloch-band edge; and for each rational slope, two line-soliton families exist. Furthermore, line solitons can bifurcate from interior points of Bloch bands as well, but such line solitons exist only for a couple of special line angles due to resonance with the Bloch bands. In addition, we show that a countable set of multiline-soliton bound states can be constructed analytically. The analytical predictions are compared with numerical results for both symmetric and asymmetric potentials, and good agreement is obtained.

A slight modification of the Kontorovich–Lebedev transform is an auto-morphism on the vector space of polynomials. The action of this -transform over certain polynomial sequences will be under discussion, and a special attention will be given to the d-orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the -transform of a 2-orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose -transform is a d-orthogonal sequence will be characterized: they are essencially semiclassical polynomials fulfilling particular conditions and d is even. The Hermite and Laguerre polynomials are the classical solutions to this problem.

Waves propagating on the surface of a three–dimensional ideal fluid of arbitrary depth bounded above by an elastic sheet that resists flexing are considered in the small amplitude modulational asymptotic limit. A Benney–Roskes–Davey–Stewartson model is derived, and we find that fully localized wavepacket solitary waves (or lumps) may bifurcate from the trivial state at the minimum of the phase speed of the problem for a range of depths. Results using a linear and two nonlinear elastic models are compared. The stability of these solitary wave solutions and the application of the BRDS equation to unsteady wave packets is also considered. The results presented may have applications to the dynamics of continuous ice sheets and their breakup.

Under investigation in this paper are the coupled Gross–Pitaevskii equations, which describe the dynamics of two-component Bose–Einstein condensates. Infinitely many conservation laws are obtained based on the Lax pair. Via the Hirota method, Bell-polynomial approach and symbolic computation, bilinear forms, Bell-polynomial-typed transformation, and bilinear-typed Bäcklund transformation are also derived. One- and two-soliton-like solutions are expressed explicitly. The gain/loss coefficient G(t) can influence the velocity of the solitonic envelopes. Head-on and overtaking elastic interactions are shown and analyzed. Inelastic interactions between two soliton-like envelopes are presented as well.

In this paper, generalizations of certain -integrals are given by the method of -difference equation, which involves the Andrews–Askey integral. In addition, some mixed generating functions for generalized Rogers–Szegö polynomials are obtained by the technique of -integral. More over, generating functions for generalized Andrews–Askey polynomials are achieved by q-integral.

A rigorous theory of the inverse scattering transform for the defocusing nonlinear Schrödinger equation with nonvanishing boundary values as is presented. The direct problem is shown to be well posed for potentials q such that , for which analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated and solved both via Marchenko integral equations, and as a Riemann-Hilbert problem in terms of a suitable uniform variable. The asymptotic behavior of the scattering data is determined and shown to ensure the linear system solving the inverse problem is well defined. Finally, the triplet method is developed as a tool to obtain explicit multisoliton solutions by solving the Marchenko integral equation via separation of variables.

The propagation of wave envelopes in two-dimensional (2-D) simple periodic lattices is studied. A discrete approximation, known as the tight-binding (TB) approximation, is employed to find the equations governing a class of nonlinear discrete envelopes in simple 2-D periodic lattices. Instead of using Wannier function analysis, the orbital approximation of Bloch modes that has been widely used in the physical literature, is employed. With this approximation the Bloch envelope dynamics associated with both simple and degenerate bands are readily studied. The governing equations are found to be discrete nonlinear Schrödinger (NLS)-type equations or coupled NLS-type systems. The coefficients of the linear part of the equations are related to the linear dispersion relation. When the envelopes vary slowly, the continuous limit of the general discrete NLS equations are effective NLS equations in moving frames. These continuous NLS equations (from discrete to continuous) also agree with those derived via a direct multiscale expansion. Rectangular and triangular lattices are examples.

By means of certain limit technique, two kinds of generalized Darboux transformations are constructed for the derivative nonlinear Schrödinger equations (DNLS). These transformations are shown to lead to two solution formulas for DNLS in terms of determinants. As applications, several different types of high-order solutions are calculated for this equation.

In this paper, we study the uniform asymptotics of the Meixner-Pollaczek polynomials with varying parameter as , where A > 0 is a constant. Two asymptotic expansions are obtained, which hold uniformly for z in two overlapping regions which together cover the whole complex plane. One involves parabolic cylinder functions, and the other is in terms of elementary functions only. Our approach is based on the steepest descent method for oscillatory Riemann-Hilbert problems first introduced by Deift and Zhou [1].

Slowly modulated water waves are considered in the presence of a strongly disordered bathymetry. Previous work is extended to the case where the random bottom irregularities are not smooth and are allowed to be of large amplitude. Through the combination of a conformal mapping and a multiple-scales asymptotic analysis it is shown that large variations of a disordered bathymetry can affect the nonlinearity coefficient of the resulting damped nonlinear Schrödinger equations. In particular it is shown that as the bathymetry fluctuation level increases the critical point (separating the focusing from the defocusing region) moves to the right, hence enlarging the region where the dynamics is of a defocusing character.

We discuss a new integrable two-component Camassa-Holm equation which describes the motion of fluid. This paper is concerned with the wave breaking mechanism for the Cauchy problem with periodic condition where two special classes of initial data are involved. Moreover, the estimate of momentum support is also shown.