Algebraic Solitons Research Articles

Algebraic solitons in the massive Thirring model.

We present exact solutions describing dynamics of two algebraic solitons in the massive Thirring model. Each algebraic soliton corresponds to a simple embedded eigenvalue in the Kaup-Newell spectral problem and attains the maximal mass among the family of solitary waves traveling with the same speed. By coalescence of speeds of the two algebraic solitons, we find a new solution for an algebraic double-soliton which corresponds to a double embedded eigenvalue. We show that the double-soliton attains the double mass of a single soliton and describes a slow interaction of two identical algebraic solitons.

Affine Algebraic Ricci Solitons Associated to the Yano Connections on Three-Dimensional Lorentzian Lie Groups

In this paper, we compute curvatures of Yano connections on three-dimensional Lorentzian Lie groups with some product structure. We define affine algebraic Ricci solitons associated to Yano connections and classify left-invariant affine algebraic Ricci solitons associated to Yano connections on three-dimensional Lorentzian Lie groups.

The two-dimensional Leznov lattice equation and its various solutions

In this paper, the two-dimensional Leznov lattice equation is studied by Hirota’s bilinear method and KP-Toda hierarchy reduction approach. We present solutions of the lattice equation in the ratio of τ functions given by the N×N Gram-type determinants. Solitons, breathers and rational solutions on constant and periodic backgrounds are derived. In particular, we find three types of breather solutions that are periodic in x-direction, y-direction and a general oblique line, respectively. Algebraic solitons are obtained by introducing two differential operators applied to the soliton solutions. We investigate the asymptotic behavior and interactions of algebraic soliton. The interaction of two algebraic solitons are analyzed and proved to be completely elastic. We demonstrate that the asymptotic algebraic solitons in the straight lines well approach the exact ones. Multiple algebraic solitons on the constant and periodic backgrounds are derived with higher-order determinant size. We introduce Schur polynomials to generate explicit expression for fundamental and higher-order lump solutions. Dynamics of lump solutions are analyzed with plots.

A special two-dimensional lattice by Blaszak and Szum: Solitons, breathers, lump solutions, and their interactions and dynamics

In this paper, we study a special two-dimensional lattice equation proposed by Blaszak and Szum. Upon utilizing the bilinear method, we derive solitons, breathers and rational solutions to the lattice equation both on the constant and periodic background. These solutions are given in terms of determinants whose matrix elements have simple algebraic expressions. In particular, we find three types of breather solutions, including Kuznetsov-Ma breather, Akhmediev breather and general one. By introducing two differential operators applied to the soliton solutions, we obtain rational solutions in terms of Schur polynomials. We demonstrate that rational solutions can exhibit algebraic solitons and lump solitons. We explore the asymptotic analysis to the algebraic solitons and find asymptotic algebraic solitons localized in the straight. By taking higher-order differential operators, we present multiple and higher-order rational solutions. The dynamical behaviors of these obtained solutions are illustrated and analyzed.

Rogue waves on the periodic background in the extended mKdV equation

We construct new exact solutions of the extended mKdV (emKdV) equation. The exact solutions are obtained by nonlinearization of spectral problem associated with the travelling periodic waves and using the one-fold, two-fold Darboux transformations. We consider the dnoidal and cnoidal travelling periodic waves of the emKdV equation. Since the dnoidal travelling periodic wave is modulationally stable, the algebraic solitons propagating on dnoidal wave background. However, since the cnoidal travelling periodic wave is modulationally unstable, the rogue waves generated on the cnoidal wave background.

Fixed points and steady solitons for the two-loop renormalization group flow

Solitons for the two-loop renormalization group flow are studied in the four-dimensional homogeneous setting, providing a classification of algebraic steady four-dimensional solitons.

Periodic waves in the discrete mKdV equation: Modulational instability and rogue waves

We derive the traveling periodic waves of the discrete modified Korteweg–de Vries equation by using a nonlinearization method. Modulational stability of the traveling periodic waves is studied from the squared eigenfunction relation and the Lax spectrum. We use numerical approximations to show that, similar to the continuous counterpart, the family of dnoidal solutions is modulationally stable and the family of cnoidal solutions is modulationally unstable. Consequently, algebraic solitons propagate on the dnoidal wave background and rogue waves (spatially and temporally localized events) are dynamically generated on the cnoidal wave background.

Algebraic Schouten solitons of Lorentzian Lie groups with Yano connections

<abstract><p>In this paper, we discuss the beingness conditions for algebraic Schouten solitons associated with Yano connections in the background of three-dimensional Lorentzian Lie groups. By transforming equations of algebraic Schouten solitons into algebraic equations, the existence conditions of solitons are found. In particular, we deduce some formulations for Yano connections and related Ricci operators. Furthermore, we find the detailed categorization for those algebraic Schouten solitons on three-dimensional Lorentzian Lie groups. The major results demonstrate that algebraic Schouten solitons related to Yano connections are present in $ G_{1} $, $ G_{2} $, $ G_{3} $, $ G_{5} $, $ G_{6} $ and $ G_{7} $, while they are not identifiable in $ G_{4} $.</p></abstract>

On the pluriclosed flow on Oeljeklaus–Toma manifolds

AbstractWe investigate the pluriclosed flow on Oeljeklaus–Toma manifolds. We parameterize left-invariant pluriclosed metrics on Oeljeklaus–Toma manifolds, and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution $\omega _t$ which once normalized collapses to a torus in the Gromov–Hausdorff sense. Moreover, the lift of $\tfrac {1}{1+t}\omega _t$ to the universal covering of the manifold converges in the Cheeger–Gromov sense to $(\mathbb H^s\times \mathbb C^s, \tilde {\omega }_{\infty })$ , where $\tilde {\omega }_{\infty }$ is an algebraic soliton.

Rational Solitons in the Gardner-Like Models

Rational solutions of nonlinear evolution equations are considered in the literature as a mathematical image of rogue waves, which are anomalously large waves that occur for a short time. In this work, bounded rational solutions of Gardner-type equations (the extended Korteweg-de Vries equation), when a nonlinear term can be represented as a sum of several terms with arbitrary powers (not necessarily integer ones), are found. It is shown that such solutions describe first-order algebraic solitons, kinks, and pyramidal and table-top solitons. Analytical solutions are obtained for the Gardner equation with two nonlinear terms, the powers of which differ by a factor of 2. In other cases, the solutions are obtained numerically. Gardner-type equations occur in the description of nonlinear wave dynamics in a fluid layer with continuous or multilayer stratification, as well as in multicomponent plasma, and their solutions are used for the interpretation of rogue waves.

Rational solutions with zero background and algebraic solitons of three derivative nonlinear Schrödinger equations: bilinear approach

Rational solutions with zero background and algebraic solitons of three derivative nonlinear Schrödinger equations: bilinear approach

Stability of Algebraic Solitons for Nonlinear Schrödinger Equations of Derivative Type: Variational Approach

We consider the following nonlinear Schr\"{o}dinger equation of derivative type: \begin{equation} i \partial_t u + \partial_x^2 u +i |u|^{2} \partial_x u +b|u|^4u=0 , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in \mathbb{R}. \end{equation} If $b=0$, this equation is a gauge equivalent form of well-known derivative nonlinear Schr\"{o}dinger (DNLS) equation. The soliton profile of the DNLS equation satisfies a certain double power elliptic equation with cubic-quintic nonlinearities. The quintic nonlinearity in the equation only affects the coefficient in front of the quintic term in the elliptic equation, so the additional nonlinearity is natural as a perturbation preserving soliton profiles of the DNLS equation. If $b>-\frac{3}{16}$, the equation has algebraic solitons as well as exponentially decaying solitons. In this paper we study stability properties of solitons by variational approach, and prove that if $b<0$, all solitons including algebraic solitons are stable in the energy space. The existence of stable algebraic solitons shows an interesting mathematical example because stable algebraic solitons are not known in the context of the corresponding double power NLS.

Instability of degenerate solitons for nonlinear Schrödinger equations with derivative

We consider the following nonlinear Schrödinger equation with derivative: (1)iut=−uxx−i|u|2ux−b|u|4u,(t,x)∈R×R,b∈R.If b=0, this equation is a gauge equivalent form of the well-known derivative nonlinear Schrödinger (DNLS) equation. The soliton profile of DNLS satisfies a certain double power elliptic equation with cubic–quintic nonlinearities. The quintic nonlinearity in (1) only affects the coefficient in front of the quintic term in the elliptic equation, so in this sense the additional nonlinearity is natural as a perturbation preserving soliton profiles of DNLS. When b≥0, Eq. (1) has degenerate solitons whose momentum and energy are zero, and if b=0, they are algebraic solitons. Inspired from the works (Martel and Merle, 2001; Farah et al., 2019) on instability theory of the L2-critical generalized KdV equation, we study the instability of degenerate solitons of (1) in a qualitative way, and when b>0, we obtain a large set of initial data yielding the instability. The arguments except one step in our proof work for the case b=0 in exactly the same way, which is a small step towards understanding the dynamics around algebraic solitons of the DNLS equation.

Perturbation of chirped localized waves in a dual-power law nonlinear medium

We investigate the transmission of localized waves through a dual-power law medium exhibiting the perturbations including the inter-modal dispersion, self-steepening, and self-frequency shift effects. A novel class of nonlinear waves that are periodic wave, kink soliton, algebraic soliton and bright soliton, along with the corresponding chirping, are reported. We found the nonlinear contribution chirp is proportional to n power of the light intensity, which is arising from the perturbations. Finally, the influence of degree of nonlinearity on the dynamical properties of the obtained chirped structures is presented.

Canonical Connections and Algebraic Ricci Solitons of Three-dimensional Lorentzian Lie Groups

In this paper, the author computes canonical connections and Kobayashi-Nomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. He defines algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. He classifies algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.

Solitons, breathers and rational solutions for a (2+1)-dimensional dispersive long wave system

In this paper, we investigate a (2+1)-dimensional dispersive long wave system with an arbitrary constant a. Solitons, line breathers, rational solutions and algebraic solitons of the system are constructed based on Hirota’s bilinear method and Kadomtsev–Petviashvili (KP) hierarchy reduction technique. These solutions are expressed in terms of N×N determinant. When the size of the determinant N is odd, solutions on periodic backgrounds are generated. Solutions on constant backgrounds are derived when N is even. By using asymptotic analysis, we elucidate explicit expressions of asymptotic algebraic solitons localized in the straight for the algebraic soliton solutions. As a byproduct, we find lump solutions for an intermediate potential of the system. The effects of constant a are discussed and it is shown that the constant can affect collision of solitons. Dynamics of the obtained solutions are discussed with plots.

Higher-order algebraic soliton solutions of the Gerdjikov–Ivanov equation: Asymptotic analysis and emergence of rogue waves

In this paper, we derive the determinant representation of higher-order algebraic soliton solutions of the Gerdjikov–Ivanov equation by using the Darboux transformation and some limit technique. Based on the asymptotic balance between different algebraic terms, we obtain the asymptotic expressions of algebraic soliton solutions with the order 2≤N≤4. It turns out that all the asymptotic solitons have the same amplitudes, most of them are located in the parabolic curves and thus have the varying velocities with the rate O(|t|−12) (except that one pair of asymptotic solitons are located in the straight lines for the odd-order cases), and they exhibit the elastic interactions of the attractive type. In addition, we reveal that the transient rogue waves are generated in the soliton-interaction region and the peak value is exactly N times the amplitude of individual soliton.

Rogue waves on the background of periodic standing waves in the derivative nonlinear Schrödinger equation.

The derivative nonlinear Schrödinger (DNLS) equation is the canonical model for the dynamics of nonlinear waves in plasma physics and optics. We study exact solutions describing rogue waves on the background of periodic standing waves in the DNLS equation. We show that the space-time localization of a rogue wave is only possible if the periodic standing wave is modulationally unstable. If the periodic standing wave is modulationally stable, the rogue wave solutions degenerate into algebraic solitons propagating along the background and interacting with the periodic standing waves. Maximal amplitudes of rogue waves are found analytically and confirmed numerically.

Potential well theory for the derivative nonlinear Schrödinger equation

We consider the following nonlinear Schr\"{o}dinger equation of derivative type: \begin{equation}i \partial_t u + \partial_x^2 u +i |u|^{2} \partial_x u +b|u|^4u=0 , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in\mathbb{R}. \end{equation} If $b=0$, this equation is known as a gauge equivalent form of well-known derivative nonlinear Schr\"{o}dinger equation (DNLS), which is mass critical and completely integrable. The equation can be considered as a generalized equation of DNLS while preserving mass criticality and Hamiltonian structure. For DNLS it is known that if the initial data $u_0\in H^1(\mathbb{R})$ satisfies the mass condition $\| u_0\|_{L^2}^2 <4\pi$, the corresponding solution is global and bounded. In this paper we first establish the mass condition on the equation for general $b\in\mathbb{R}$, which is exactly corresponding to $4\pi$-mass condition for DNLS, and then characterize it from the viewpoint of potential well theory. We see that the mass threshold value gives the turning point in the structure of potential wells generated by solitons. In particular, our results for DNLS give a characterization of both $4\pi$-mass condition and algebraic solitons.

Double Extensions on Riemannian Ricci Nilsolitons

In this paper, we study left invariant Lorentz algebraic Ricci solitons on nilpotent Lie groups. Based on the concept of Lorentz datum and the technique of double extensions, we are able to construct Lorentz Ricci nilsolitons from any Riemannian Ricci nilsoliton. Conversely, we prove that any Lorentz Ricci nilsoliton with degenerate center is a double extension of a Riemannian Ricci nilsoliton with respect to a Lorentz data. Moreover, we provide a strategy to classify Lorentz Ricci nilsolitons with degenerate center. As an application, we present a large number of Lorentz Ricci nilsolitons based on Abelian Riemannian Ricci solitons. Finally, a family of left invariant Lorentz Ricci nilsolitons on a 9-dimensional nilpotent Lie algebra is given.