Nonlinear Sciences

A nonlocal Alice-Bob Kadomtsev-Petviashivili (ABKP) system with shifted-parities ( P ^ x s and P ^ y s parities with shifts for the space variables x and y ) and delayed time reversal ( T ^ d , time reversal with a delay) symmetries is investigated. Some types of P ^ y s P ^ x s T ^ d invariant solutions including multiple soliton solutions, Painlevé reductions and soliton and p -wave interaction solutions are obtained via P ^ y s P ^ x s T ^ d symmetry and the solutions of the usual local KP equation. Some special P ^ y s P ^ x s T ^ d symmetry breaking multi-soliton solutions and cnoidal wave solutions are found from the P ^ y s P ^ x s T ^ d symmetry reduction of a coupled local KP system.

We provide closed form solutions for an equation which describes the transport of turbulent kinetic energy in the framework of a turbulence model with a single equation.

Recently, finding exact solutions of nonlinear fractional differential equations has attracted great interest. In this paper, the space time-fractional Klein-Gordon equation with cubic nonlinearities is examined. Firstly, suitable exact soliton solutions are formally extacted by using the solitary wave ansatz method. Some solutions are also illustrated by the computer simulations. Besides, the modified Kudryashov method is used to construct exact solutions of this equation.

In this article, we study the generalised Kudryashov method for the time fractional generalized Burgers-Fisher equation (GBF). Using traveling wave transformation, the time fractional GBF is transformed to nonlinear ordinary differential equation (ODE). Later, in the nonlinear ODE of timefractional GBF, the generalized Kudryashov and power series method is applied to get exact solutions.

In this paper, we firstly construct a weakly coupled Toda lattices with indefinite metrics which consist of 2N different coupled Hamiltonian systems. Afterwards, we consider the iso-spectral manifolds of extended tridiagonal Hessenberg matrix with indefinite metrics what is an extension of a strict tridiagonal matrix with indefinite metrics. For the initial value problem of the extended symmetric Toda hierarchy with indefinite metrics, we introduce the inverse scattering procedure in terms of eigenvalues by using the Kodama's method. In this article, according to the orthogonalization procedure of Szegö, the relationship between the τ -function and the given Lax matrix is also discussed. We can verify the results derived from the orthogonalization procedure with a simple example. After that, we construct a strongly coupled Toda lattices with indefinite metrics and derive its tau structures. At last, we generalize the weakly coupled Toda lattices with indefinite metrics to the Z n -Toda lattices with indefinite metrics.

We study the fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation in the limit of large order. Using a recently-proposed Riemann-Hilbert representation of the rogue wave solution of arbitrary order k , we establish the existence of a limiting profile of the rogue wave in the large- k limit when the solution is viewed in appropriate rescaled variables capturing the near-field region where the solution has the largest amplitude. The limiting profile is a new particular solution of the focusing nonlinear Schrödinger equation in the rescaled variables --- the rogue wave of infinite order --- which also satisfies ordinary differential equations with respect to space and time. The spatial differential equations are identified with certain members of the Painlevé-III hierarchy. We compute the far-field asymptotic behavior of the near-field limit solution and compare the asymptotic formulæ\ with the exact solution with the help of numerical methods for solving Riemann-Hilbert problems. In a certain transitional region for the asymptotics the near field limit function is described by a specific globally-defined tritronquée solution of the Painlevé-II equation. These properties lead us to regard the rogue wave of infinite order as a new special function.

We obtain closed-form solutions of several inhomogeneous Lienard equations by the factorization method. The two factorization conditions involved in the method are turned into a system of first-order differential equations containing the forcing term. In this way, one can find the forcing terms that lead to integrable cases. Because of the reduction of order feature of factorization, the solutions are simultaneously solutions of first-order differential equations with polynomial nonlinearities. The illustrative examples of Lienard solutions obtained in this way generically have rational parts, and consequently display singularities.

We address the classical factorization problem of a one dimensional Schrödinger operator − ∂ 2 +u−λ , for a stationary potential u of the KdV hierarchy but, in this occasion, a "parameter" λ . Inspired by the more effective approach of Gesztesy and Holden to the "direct" spectral problem, we give a symbolic algorithm by means of differential elimination tools to achieve the aimed factorization. Differential resultants are used for computing spectral curves, and differential subresultants to obtain the first order common factor. To make our method fully effective, we design a symbolic algorithm to compute the integration constants of the KdV hierarchy, in the case of KdV potentials that become rational under a Hamiltonian change of variable. Explicit computations are carried for Schrödinger operators with solitonic potentials.

After characterizing the integrable discrete analogue of the Euler's elastica, we focus our attention on the problem of approximating a given discrete planar curve by an appropriate discrete Euler's elastica. We carry out the fairing process via a L 2 -distance minimization to avoid the numerical instabilities. The optimization problem is solved via a gradient-driven optimization method (IPOPT). This problem is non-convex and the result strongly depends on the initial guess, so that we use a discrete analogue of the algorithm provided by Brander et al., which gives an initial guess to the optimization method.

In this work, we employ the algebraic-differential method recently developed by the author to solve the Yang-Baxter equation for arbitrary fifteen-vertex models satisfying the ice-rule. We show that there are four different families of such regular R matrices containing several free-parameters. The corresponding reflection K matrices, solutions of the boundary Yang-Baxter equation, were also found and classified. We found that there are three different families of regular K matrices, regardless of what R matrix we choose.