The algebraic structures of integrable hierarchies play an important role in the study of soliton equations. In this paper, we use splitting theory to give a matrix representation of a constrained CKP hierarchy, which can be considered as a generalization of the A(2n)(<^>(2))-KdV hierarchy and the constrained KP hierarchy. An equivalent construction in terms of the pseudo-differential operator is discussed. Darboux transformations, scaling transformations, and tau functions for this constrained hierarchy are studied.

We explore the properties of the derivative nonlinear Schrödinger-type systems that are associated to irreducible compact Hermitian symmetric spaces. We derive the derivative nonlinear Schrödinger systems using a consistent algebraic framework of Loop group factorization the give a unified construction of Darboux transformations for these derivative nonlinear Schrödinger-type systems.

We develop a unified algebraic framework for nonlocal derivative nonlinear Schrödinger (DNLS)-type hierarchies based on loop algebra splittings. Within this framework, nonlocal and reverse space-time reductions are realized through algebraic constraints, leading to the corresponding integrable hierarchies. By constructing simple elements and establishing the associated factorization theory, we derive Darboux transformations (DTs). And apply DTs to construct explicit solutions.

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

The Â_2n^(2) -hierarchy can be constructed from a splitting of the Kac–Moody algebra of type Â_2n^(1) by an involution. By choosing certain cross section of the gauge action, we obtain the Â_2n^(2) -KdV hierarchy. They are the equations for geometric invariants of isotropic curve flows of type A, which gives a geometric interpretation of the soliton hierarchy. In this paper, we construct Darboux and Bäcklund transformations for the Â_2n^(2) -hierarchy, and use it the construct Darboux transformations for the Â_2n^(2) -KdV hierarchy and isotropic curve flows of type A. Moreover, explicit soliton solutions for these hierarchies are given.

The $$\hat{A}_{2n}^{(2)}$$ -hierarchy can be constructed from a splitting of the Kac–Moody algebra of type $$\hat{A}_{2n}^{(1)}$$ by an involution. By choosing certain cross section of the gauge action, we obtain the $$\hat{A}_{2n}^{(2)}$$ -KdV hierarchy. They are the equations for geometric invariants of isotropic curve flows of type A, which gives a geometric interpretation of the soliton hierarchy. In this paper, we construct Darboux and Bäcklund transformations for the $$\hat{A}_{2n}^{(2)}$$ -hierarchy, and use it the construct Darboux transformations for the $$\hat{A}_{2n}^{(2)}$$ -KdV hierarchy and isotropic curve flows of type A. Moreover, explicit soliton solutions for these hierarchies are given.

The ˆ A ( 2 ) 2 n -hierarchy can be constructed from a splitting of the Kac–Moody algebra of type ˆ A ( 1 ) 2 n by an involution. By choosing certain cross section of the gauge action, we obtain the ˆ A ( 2 ) 2 n -KdV hierarchy. They are the equations for geometric invariants of isotropic curve flows of type A, which gives a geometric interpretation of the soliton hierarchy. In this paper, we construct Darboux and Bäcklund transformations for the ˆ A ( 2 ) 2 n -hierarchy, and use it the construct Darboux transformations for the ˆ A ( 2 ) 2 n -KdV hierarchy and isotropic curve flows of type A. Moreover, explicit soliton solutions for these hierarchies are given.

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

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

In this paper, the modified KP hierarchy in the Kupershmidt–Kiso version is extended to the super case by Kac–van de Leur construction, that is, using highest weight representations of the even part in the tensor product of the infinite-dimensional Lie superalgebra gl∞|∞ with Grassmann algebra G. First, the super modified KP (SmKP) hierarchy is constructed in terms of superfermionic bilinear equations. Then, the superbosonic form of the SmKP hierarchy is given by super boson–fermion correspondence. With the help of super Hirota bilinear operators, the corresponding super Hirota bilinear equations of the SmKP hierarchy are obtained. Next, the Darboux transformations of this new SmKP hierarchy are expressed in the form of free superfermions and various solutions are derived. Finally, the super bilinear equations in the form of super wave functions are also constructed from the superbosonic ones, which is hoped to be helpful to obtain the corresponding Lax structures.

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.

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

This paper is a survey on the relationship between the invariant geometric curve flows and the soliton equations.First,we show that the classical integrable equations including the KdV equation,the modified KdV equation,the nonlinear Schr(o)dinger equation and the Burgers equation arise naturally from the invariant planar or space curve flows in centro-equiaffine,Euclidean and similarity geometries.The curve flows corresponding to some special solutions of corresponding integrable equations are identified.Second,the geometric formula-tion to typical properties of integrable systems such as the Miura transformation,B(a)cklund transformation and bi-Hamiltonian structure are provided.Finally,the geometric curve flows corresponding to the Camassa-Holm-type equations are presented.

A smooth curve gamma in Rn+1, n is isotropic if gamma, gamma(x),..., gamma((2 n))(x) are linearly independent and the span of gamma, gamma(x),..., gamma((n-1))(x) is isotropic. We construct two hierarchies of isotropic curve flows on R-n+1,R-n, whose differential invariants are solutions of Drinfeld-Sokolov's KdV type soliton hierarchies associated to the affine Kac-Moody algebra (B) over cap ((1))(n) and (A) over cap ((2))(2n). For example, the (B) over cap ((1))(1)-KdV is the KdV hierarchy and the (A) over cap ((2))(2)-KdV hierarchy is the Kupershmidt-Kaup (KK) hierarchy. Hence we our study gives geometric interpretations of the KdV and KK equations as the curvature flows of natural geometric curve flows on the light cone of R-2,R-1. Bi-Hamiltonian structures and conservation laws for isotropic curve flows on R-n+1,R-n are also given.

The $${\hat{B}}_n^{(1)}$$ -hierarchy is constructed from the standard splitting of the affine Kac–Moody algebra $${\hat{B}}_n^{(1)}$$ , the Drinfeld–Sokolov $${\hat{B}}_n^{(1)}$$ –KdV hierarchy is obtained by pushing down the $${\hat{B}}_n^{(1)}$$ -flows along certain gauge orbit to a cross section of the gauge action. In this paper, we

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

Series of deformed Camassa-Holm-type equations are constructed using the Lagrangian deformation and Loop algebra splittings. They are weakly integrable in the sense of modified Lax pairs.

The n-fold Darboux transformation \(T_{n}\) of the focusing real modified Korteweg–de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the n-soliton solutions of the mKdV equation are also expressed by determinants whose elements consist of the eigenvalues \(\lambda _{j}\) and the corresponding eigenfunctions of the associated Lax equation. The nonsingular n-positon solutions of the focusing mKdV equation are obtained in the special limit \(\lambda _{j}\rightarrow \lambda _{1}\), from the corresponding n-soliton solutions and by using the associated higher-order Taylor expansion. Furthermore, the decomposition method of the n-positon solution into n single-soliton solutions, the trajectories, and the corresponding “phase shifts” of the multi-positons are also investigated.

We construct Bäcklund transformations (BT) for the Gelfand-Dickey hierarchy (GD_n-hierarchy) on the space of n-th order differential operators on the line. Suppose L=∂_x^n-∑_i=1^n-1u_i∂_x^(i-1) is a solution of the j-th GD_n flow. We prove the following results: (1) There exists a system (BT)_u,k of non-linear ordinary differential equations for h:R^2→ C depending on u_1, …, u_n-1 in x and t variables such that L̃= (∂+h)^-1L(∂+h) is a solution of the j-th GD_n flow if and only if h is a solution of (BT)_u,k for some parameter k. Moreover, coefficients of L̃ are differential polynomials of u and h. We say such L̃ is obtained from a BT with parameter k from L. (2) (BT)_u,k is solvable. (3) There exists a compatible linear system for ϕ:R^2→ C depending on a parameter k, such that if ϕ_1, …, ϕ_n-1 are linearly independent solutions of this linear system then h:=(ln W(ϕ_1, …, ϕ_n-1))_x is a solution of (BT)_u,k and (∂+h)^-1 L (∂+h) is a solution of the j-th GD_n flow, where W(ϕ_1,…,ϕ_n-1) is the Wronskian Moreover, these give all solutions of (BT)_u,k. (4) We show that the BT for the GD_n hierarchy constructed by M. Adler is our BT with parameter k=0. (5) We construct a permutability formula for our BTs and infinitely many families of explicit rational solutions and soliton solutions.

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