相关论文: A new Evans function for quasi-periodic solutions …

Two concepts, very different in nature, have proved to be useful in analytical and numerical studies of spectral stability: (i) the Krein signature of an eigenvalue, a quantity usually defined in terms of the relative orientation of certain…

The sine(sinh)-Gordon hierarchy of integrable Hamiltonian systems is described in detail, and all dynamic variables are expressed in terms of the $\wp$-functions that uniformize the associated spectral curve. Quasi-periodic solutions to the…

We demonstrate a geometrically inspired technique for computing Evans functions for the linearised operators about travelling waves. Using the examples of the F-KPP equation and a Keller-Segel model of bacterial chemotaxis, we produce an…

This paper presents a comprehensive analysis of several aspects of the sinh-Gordon equation within a periodic setting. Our investigation proceeds in three main stages. First we establish the existence of periodic solutions for a fixed wave…

We give an exhaustive, non-perturbative classification of exact travelling-wave solutions of a perturbed sine-Gordon equation (on the real line or on the circle) which is used to describe the Josephson effect in the theory of…

We present a set of time quasi-periodic solutions to a nonlinear Klein-Gordon equation with a decaying nonlinear term on the torus in arbitrary dimensions. This paper generalizes the bifurcation method developed in [W2].

The Evans function is an analytic function that encodes information about the intersection of certain subspaces in ODE boundary value problems. As such it is a useful tool for computing the spectrum of boundary value problems arising in the…

The Klein-Gordon equation in the presence of a strong electric field, taking the form of the Mathieu equation, is studied. A novel analytical solution is derived for particles whose asymptotic energy is much lower or much higher than the…

The Evans function has been used extensively to study spectral stability of travelling-wave solutions in spatially extended partial differential equations. To compute Evans functions numerically, several shooting methods have been…

In this work, we systematically generalize the Evans function methodology to address vector systems of discrete equations. We physically motivate and mathematically use as our case example a vector form of the discrete nonlinear Schrodinger…

We establish instability of periodic traveling waves arising in conservation laws featuring phase transition. The analysis uses the Evans function framework introduced by R.A. Gardner in the periodic case. The main new tool is a periodic…

In the paper we present a functional-discrete method for solving the Goursat problem for nonlinear Klein-Gordon equation. The sufficient conditions providing that the proposed method converges superexponentially are obtained. The results of…

In this paper we discuss some exact results related to the fractional Klein--Gordon equation involving fractional powers of the D'Alembert operator. By means of a space-time transformation, we reduce the fractional Klein--Gordon equation to…

We study the point spectrum of the linearisation of Euler's equation for the ideal fluid on the torus about a shear flow. By separation of variables the problem is reduced to the spectral theory of a complex Hill's equation. Using Hill's…

The solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann-Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are…

The concept of parity due to Fitzpatrick, Pejsachowicz and Rabier is a central tool in the abstract bifurcation theory of nonlinear Fredholm operators. In this paper, we relate the parity to the Evans function, which is widely used in the…

In the spectral stability analysis of localized patterns to singular perturbed evolution problems, one often encounters that the Evans function respects the scale separation. In such cases the Evans function of the full linear stability…

We present a numerical study of spatially quasi-periodic traveling waves on the surface of an ideal fluid of infinite depth. This is a generalization of the classic Wilton ripple problem to the case when the ratio of wave numbers satisfying…

This work aims to initiate a discussion on finding solutions to non-homoge\-neous differential equations in terms of generalized functions. For simplicity, we conduct the analysis within the specific context of the stationary Klein-Gordon…

We propose a procedure for computing the direct scattering transform of the periodic sine-Gordon equation. This procedure, previously used within the periodic Korteweg-de Vries equation framework, is implemented for the case of the…