We construct higher order rogue wave solutions for the Gerdjikov-Ivanov equation explicitly in term of determinant expression.Dynamics both soliton and non-soliton is discussed.A family with distinct structures are presented, which new to equation.

The integrable nonlocal Lakshmanan—Porsezian—Daniel (LPD) equation which has the higher-order terms (dispersions and nonlinear effects) is first introduced. We demonstrate integrability of LPD equation, provide its Lax pair, present rational soliton solutions self-potential function by using degenerate Darboux transformation. From numerical plots solutions, compression effects real refractive index profile gain-or-loss distribution produced δ are discussed.

China is facing development challenges, such as the red line of arable land, resource shortage, and tightening ecological environmental constraints. In this context, improving land green utilization efficiency (LGUE) not only an important undertaking to optimize spatial layout country improve carrying capacity but also inevitable choice for comprehensive transformation economic social development. China’s energy-consuming right trading (ECRT) energy transition demonstration policy; however,...

A special chain of the gauge transformations two-component constrained KP is introduced, and then transformed components τ-function this hierarchy are presented. Based on this, by different reductions, BKP CKP hierarchies obtained. Furthermore, component functions τ-functions them expressed determinants with help determinant representation transformation operators.

In this paper, the modified KP hierarchy in Kupershmidt–Kiso version is extended to super case by Kac–van de Leur construction, that is, using highest weight representations of even part tensor product infinite-dimensional Lie superalgebra gl∞|∞ with Grassmann algebra G. First, (SmKP) constructed terms superfermionic bilinear equations. Then, superbosonic form SmKP given boson–fermion correspondence. With help Hirota operators, corresponding equations are obtained. Next, Darboux...

For $n$-dimensional central affine curve flows, we 1) solve the Cauchy problem with periodic initial data and having rapidly decaying curvatures, 2) construct Bäcklund transformations, a Permutability formula, explicit solutions, 3) write down formulas for Bi-Hamiltonian structure conservation laws.

We generate hierarchies of derivative nonlinear Schrödinger-type equations and their nonlocal extensions from Lie algebra splittings automorphisms. This provides an algebraic explanation some known reductions newly established in integrable systems.

We construct Bäcklund transformations (BT) for the Gelfand–Dickey hierarchy (GD|$_n$|-hierarchy) on space of |$n$|th order differential operators line. Suppose |$L = \partial_x^n - \sum_{i=1}^{n=1}\, u_i \partial_x^{i-1}$| is a solution |$j$|th GD|$_n$| flow. prove following results: (1) There exists system (BT)|$_{u,k}$| non-linear ordinary equations |$h : R^2\to C$| depending |$u_1, \ldots , u_{n-1}$| in |$x$| and |$t$| variables such that |$\tilde{L} (\partial + h)^{-1}L(\partial h)$|...

In this paper, we provide a simple method to generate higher order position solutions and rogue wave for the derivative nonlinear Schr\"odinger equation. The formulae of these are given in terms determinants. dynamics structures generated by studied.

We construct a sequence of commuting central affine curve flows on $R^n\backslash 0$ invariant under the action $SL(n,R)$ and prove following results: (a) The curvatures solution j-th flow is Gelfand-Dickey (GD$_n$) hierarchy space n-th order differential operators. (b) use Cauchy problems GD$_n$ to solve for with periodic initial data also whose are rapidly decaying. (c) obtain bi-Hamiltonian structure that it arises naturally from Poisson structures certain co-adjoint orbits. (d) Backlund...

The Miura links between the KP and modified hierarchies are extended to SUSY (SKP) (SmKP) of Manin Radul Jacobian types. corresponding Darboux transformations in case constructed by using links. In this sense, one can better understand why step transformation SKP hierarchy not keep odd flows. addition, SmKP be obtained naturally Then squared eigenfunction symmetry is link from hierarchy. At last, starting results hierarchy, constrained type also link, that is, adding appropriate flow parts. discussed

Abstract The Miura transformation plays a crucial role in the study of integrable systems. There have been various extensions transformation, which used to relate different kinds equations and classify bi‐Hamiltonian structures. In this paper, we are mainly concerned with geometric aspects transformation. generalized transformations from mKdV‐type hierarchies KdV‐type constructed under both algebraic settings. It is shown that not only curve flows geometries but also induce transition...

We use loop group factorizations to construct Darboux transformations, Permutability formulas, Scaling Transformations, and explicit soliton solutions of the [Formula: see text]-d Schrödinger flows on compact Hermitian symmetric spaces.

Let $R^{n+1, n}$ be the vector space $R^{2n+1}$ equipped with bilinear form $(X,Y)=X^t C_n Y$ of index $n$, where $C_n= \sum_{i=1}^{2n+1} (-1)^{n+i-1} e_{i, 2n+2-i}$. A smooth $γ: R\to R^{n+1,n}$ is {\it isotropic} if $γ, γ_x, \ldots, γ_x^{(2n)}$ are linearly independent and span γ_x^{(n-1)}$ isotropic. Given an isotropic curve, we show that there a unique up to translation parameter such $(γ_x^{(n)}, γ_x^{(n)})=1$ (we call parameter) also exists natural moving frame. In this paper, consider...

A simple decoupling method for multiple patch antennas is presented. The proposed 1×2 are placed side by with zero spacing. By inserting metallic vias, the coupled separated electromagnetically, although they physically together. isolation between two ports can be enhanced, together restored radiation patterns. To verify idea, 1 × 2 designed, fabricated and measured. Good agreement observed measured simulated results. overlapped impedance bandwidths 1.34% larger than 18 dB. It should...

A smooth map γ in the symplectic space R2n is Lagrangian if γ,γx,…, γx(2n−1) are linearly independent and span of γ,γx,…,γx(n−1) a subspace R2n. In this paper, we (i) construct complete set differential invariants for curves with respect to group Sp(2n), (ii) two hierarchies commuting Hamiltonian curve flows C-type A-type, (iii) show that solutions A-type Drinfeld-Sokolov’s C^n(1)-KdV A^2n−1(2)-KdV respectively, (iv) Darboux transforms, Permutability formulas, scaling give an algorithm...