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Molecular adsorption conformations and arrangement configurations on surfaces are important structural aspects of surface stereochemistry, but their roles in steering the structures of metal-organic networks (MONs) remain vague and unexplored. In this study, we constructed MONs by the coordination self-assembly of isocyanides on Cu(111) and Ag(111) surfaces and demonstrated that the MON structures can be steered by surface stereochemistry, including the adsorption conformations of the isocyanide molecules and the arrangement configurations of the coordination nodes and subunits. The coordination self-assembly of 1,4-phenylene diisocyanobenzene afforded a honeycomb MON consisting of 3-fold (isocyano)3-Cu motifs on a Cu(111) surface. In contrast, geometrically different chevron-shaped 1,3-phenylene diisocyanobenzene (m-DICB) failed to generate a MON, which is ascribable to its standing conformation on the Cu(111) surface. However, m-DICB was adsorbed in a flat conformation on a Ag(111) surface, which has a larger lattice constant than a Cu(111) surface, and smoothly underwent coordination self-assembly to form a MON consisting of (isocyano)3-Ag motifs. Interestingly, only C3-Ag nodes with heterotactic configurations could grow into larger subunits; those subunits with heterotactic configurations further grew into Sierpinski triangle fractals (up to fourth order), while subunits with homotactic configurations afforded a triangular MON.

Dendrimers are homostructural and highly branched macromolecules with unique dendritic effects and extensive use in multidisciplinary fields. Although thousands of dendrimers have been synthesized in solution, the on-surface synthetic protocol for planar dendrimers has never been explored, limiting the elucidation of the mechanism of dendritic effects at the single-molecule level. Herein, we describe an on-surface synthetic approach to planar dendrimers, in which exogenous palladium is used as a catalyst to address the divergent cross-coupling of aryl bromides with isocyanides. This reaction enables one aryl bromide to react with two isocyanides in sequential steps to generate the divergently grown product composed of a core and two branches with high selectivity and reactivity. Then, a dendron with four branches and dendrimers with eight or twelve branches in the outermost shell are synthesized on Au(111). This work opens the door for the on-surface synthesis of various planar dendrimers and relevant macromolecular systems., Although many strategies exist to synthesize dendrimers in solution, the synthesis of planar dendrimers on a surface has proven challenging. Here, the authors produce planar dendrimers through a divergent on-surface cross-coupling reaction between one aryl bromide and two isocyanides, which enables the growth of branches from a single reactive site.

The non-isospectral Ablowitz–Ladik hierarchy is integrated by the inverse scattering transform. In contrast with the isospectral Ablowitz–Ladik hierarchy, the eigenvalues of the non-isospectral Ablowitz–Ladik equations in the scattering data are time-dependent. The multi-soliton solution for the hierarchy is presented. The reductions to the non-isospectral discrete NLS hierarchy and the non-isospectral discrete mKdV hierarchy and their solutions are considered.

The generalized double Wronskian solutions of the third-order isospectral AKNS equation are obtained. Thus we found rational solutions, Matveev solutions, complexitons and interaction solutions. Moreover, rational solutions of the mKdV equation and KdV equation in double Wronskian form are constructed by reduction.

The double Wronskian solutions whose entries satisfy matrix equation of the general nonlinear Schrodinger equation with derivative (GDNLSE) are derived through the Wronskian technique. Soliton solutions and rational solutions of GDNLSE are obtained by taking special cases in general solutions.

First, a new multi-component modified Kadomtsev–Petviashvill (mKP) spectral problem is constructed by k-constraint imposed on a general pseudo-differential operator. Then, two hierarchies of multi-component mKP equations are derived, including positive non-isospectral mKP hierarchy and negative non-isospectral mKP hierarchy. Moreover, new integrable couplings of the resulting mKP soliton hierarchies are constructed by enlarging the associated matrix spectral problem.

The generalized ( 2 + 1 ) -dimensional KP, cKP and mKP are decomposed into the known ( 1 + 1 ) -dimensional soliton equations. Then, we show that the ( 1 + 1 ) -dimensional soliton equations give rise to the explicit soliton solutions of the generalized KP, cKP and mKP.

A nonisospectral ( 2 + 1 )-dimensional breaking soliton equation (( 2 + 1 )DBSE) is derived, which corresponds to the spectral parameter λ satisfying 2 λ λ y − λ t = λ 2 . The bilinear form for the nonisospectral ( 2 + 1 )DBSE is obtained and the multisoliton solutions are worked out by means of the Hirota method and Wronskian technique, respectively. The nonisospectral ( 2 + 1 )-dimensional breaking nonlinear Schrodinger equation and its multisoliton solutions are also presented by reduction.

The relations between the consistent solutions to the first two equations of the AKNS hierarchy and the solutions for three ( 2 + 1 )-dimension equations are discussed. Making use of the theory of ( 1 + 1 )-dimension soliton equations and Wronskian technique, soliton solutions, rational solutions, Matveev solutions, complexitons and mixed solutions of the three ( 2 + 1 )-dimension equations are obtained. Specially, the lump-like solutions of the GKPI equation and CKP equation, the periodic solutions of the GKPI equation and the GMKPI equation are also obtained. Besides, the dynamics including one-solitons characteristics, resonances of the two-solitons related to the GKPI equation are investigated analytically.

Gauge transform between the first order nonisospectral AKNS spectral problem and isospectral AKNS spectral problem is obtained, corresponding transform operator between the first order nonisospectral AKNS hierarchy and isospectral AKNS hierarchy is given.

A practicable way to construct discrete integrable couplings is proposed by making use of two types of semi-direct sum Lie algebras. As its application, two kinds of discrete integrable couplings of the Volterra lattice are worked out.

A hierarchy of nonlinear differential-difference equations is derived from a new discrete spectral problem. The Hamiltonian structure of the resulting hierarchy is constructed by means of a trace identity formula. Moreover, a new Bargmann type integrable system associated with the hierarchy is presented by applying the binary nonlinearization approach of Lax pairs. Based on the symmetry constraints and the generating function of integrals of motion, the resulting system is further proved to be completely integrable Hamiltonian system in the Liouville sense.

A new representation of N -soliton solution of the fifth order KdV equation is obtained by using Backlund transformation method. We show a direct method for constructing the novel N -soliton solution by performing an appropriate limiting procedure on the known soliton solutions.

A new spectral problem and the associated integrable hierarchy of nonlinear evolution equations are presented in this paper. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. An explicit symmetry constraint is proposed for the Lax pairs and the adjoint Lax pairs of the hierarchy. Moreover, the corresponding Lax pairs and adjoint Lax pairs are nonlinearized into a hierarchy of commutative, new finite-dimensional completely integrable Hamiltonian systems in the Liouville sense. Further, an involutive representation of solution of each equation in the hierarchy is given. Finally, expanding integrable models of the hierarchy are constructed by using a new Loop algebra.

In terms of properties of the known loop algebra A ∼ 1 and difference operators, a new algebraic system χ is constructed. By using the algebraic system χ, a discrete matrix spectral problem is introduced and a hierarchy of nonlinear lattice equations is derived. From the hierarchy the celebrated cubic Volterra lattice equation is engendered. We call the hierarchy a generalized cubic Volterra hierarchy. Then an extended algebraic system χ ˜ of χ is presented, from which the integrable couplings system of the generalized cubic Volterra lattice are obtained.

The Lax pair of the mixed Ablowitz–Kap–Newell–Segur (AKNS) system is obtained from compatibility condition. Hirota's bilinear form is derived by some dependent variable transformation. Moreover, by means of the Wronskian technique, the double Wronskian form of soliton solutions are found. Specially, the two-soliton solution is presented.

The bilinear equation of the non-isospectral AKNS equation is derived. The N -soliton solutions and the double Wronskian solution are obtained through the Hirota method and the Wronskian technique, respectively. A non-isospectral Schrodinger equation and its multi-soliton solutions are given by reducing.

In this Letter, a systematic method is presented to construct the Darboux transformation with multi-parameters for the coupled nonlinear Schrodinger equations, from which the solutions of Schrodinger equations are reduced to solving a linear algebraic system and a constraint differential equation. With the aid of symbolic computation, new soliton-like solutions for the nonlinear Schrodinger equations are obtained by using its Darboux transformation.

A bilinear form of a nonisospectral sine-Gordon equation is given. Exact solutions are further obtained through the Hirota method and Wronskian technique, respectively. Some nonisospectral characteristics of the obtained solutions, such as the time-dependent shape, time-dependent speed and some non-propagating solitary waves, are discussed. The conservation laws are derived.

A new simple loop algebra G ∼ M is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component WKI hierarchy is obtained. Furthermore, an expanding loop algebra F ∼ M of the loop algebra G ∼ M is presented. Based on F ∼ M , the multi-component integrable couplings system with two arbitrary functions of the multi-component WKI hierarchy is worked out. How to design the isospectral problem of the multi-component hierarchy of equations is a technique and interesting topic. The method can be applied to other nonlinear evolution equations hierarchy.

It is shown that a modified Backlund transformation is presented by the dependent variable transformation. Starting from it, a new representation of N -soliton solution and a class of novel multi-soliton solution of the Boussinesq equation have been derived. We also find the novel soliton solution may be deduced by limiting.

In this Letter, a new isospectral problem is established by choosing subalgebra of loop algebra A 2 as its basis. By making use of Tu's scheme, the N Hamiltonian structure is established and it is shown that the new hierarchy is integrable in Liouville sense. As two reductions, the two generalized nonlinear Schrodinger equations are obtained, respectively. Integrable couplings of the new hierarchy are obtained by taking a proper algebraic transformation to turn the loop algebra A 2 into its equivalent loop algebra like the form of A 1 . Finally, the Hamiltonian form of a binary symmetric constrained flows of the system is given.

Using homogenous balance method we obtain Backlund transformation (BT) and a linear partial differential equation of (2+1)-dimensional dispersive long water equations (DLWE). As a result, multisoliton and single soliton and other exact solutions of (2+1)-dimensional DLWE are obtained. By analyzing single soliton solution, we get some dromions solutions. This method, which can be generalized to some (2+1)-dimensional nonlinear evolution equations, is simple and powerful.

An eigenvalue problem and the associated new Lax integrable hierarchy of nonlinear evolution equations are presented in this paper. As two reductions, the generalized nonlinear Schrodinger equations and the generalized mKdV equations are obtained. Zero curvature representation and bi-Hamiltonian structure are established for the whole hierarchy based on a pair of Hamiltonian operators (Lenard's operators), and it is shown that the hierarchy of nonlinear evolution equations is integrable in Liouville's sense. Thus the hierarchy of nonlinear evolution equations has infinitely many commuting symmetries and conservation laws. Moreover the eigenvalue problem is nonlinearized as a finite-dimensional completely integrable system under the Bargmann constraint between the potentials and the eigenvalue functions. Finally finite-dimensional Liouville integrable system are found, and the involutive solutions of the hierarchy of equations are given. In particular, the involutive solutions are developed for the system of generalized nonlinear Schrodinger equations.

Soliton-like solutions for a nonisospectral KdV hierarchy are obtained by using the inverse scattering transform. The interaction of two solitons is investigated.

A new bilinear Backlund transformation is presented for the shallow water waves equation. Starting from the new Backlund transformation, a wide variety of novel soliton solutions of the shallow water waves equation are generated. Moreover, the relationship between these solutions has been discussed.

Solitons, negatons, positons, rational-like solutions and mixed solutions of a non-isospectral equation, the Korteweg–de Vries equation with loss and non-uniformity terms, are obtained through the Wronskian technique. The non-isospectral characteristics of the motion behaviours of some solutions are described with some figures made by using Mathematica. We also derive an infinite number of conservation laws of the equation.

The KP equation with self-consistent sources is derived through the linear problem of the KP system. The multisoliton solutions for the KP equation with self-consistent sources are obtained by using Hirota method and Wronskian technique. The coincidence of these solutions is shown by direct computation.

The paper proposes an approach to constructing the symmetries and their algebraic structures for isospectral and nonisospectral evolution equations of (1+2)-dimensional systems associated with the linear problem of Sato theory. To do that, we introduce the implicit representations of the isospectral flows {Km} and nonisospectral flows {σn} in the high dimensional cases. Three examples, the Kodomstev–Petviashvili system, BKP system and new CKP system, are considered to demonstrate our method.

A systematic approach to constructing an infinite number of conservation laws for discrete soliton systems is proposed, and three examples are given.

We describe an approach for investigating the Hamiltonian structures of the lattice isospectral evolution equations associated with a general discrete spectral problem. By using the so-called implicit representations of the isospectral flows, we demonstrate the existence of the recursion operator L, which is a strong and hereditary symmetry of the flows. It is then proven that every equation in the isospectral hierarchy possesses the Hamiltonian structure if L has a skew-symmetric factorization and the first equation (ut = K(0)) in the hierarchy satisfies some simple condition. We obtain related properties, such as the implectic-symplectic factorization of L, the Liouville complete integrability and the multi-Hamiltonian structures of the isospectral hierarchy. Four examples are given.

The nonisospectral Kadomtshev–Petviashvili (KP) equation is solved by the Hirota method and Wronskian technique. Exact solutions that possess soliton characters with nonisospectral properties are obtained. In addition, rational and mixed solutions are derived. We also obtain a new molecular equation that admits a solution in the Wronskian form.

In the division of time imaging polarimetry, polarization information of field under measurement (I, Q, U), is obtained by rotating the analyzer. In the process of measurement, the beam deviation caused by the wedge of the analyzer reduces the spatial resolution and polarization accuracy of imaging polarimetry. In this paper we present that the beam deviation above can be reduced or even compensated for by adjusting the tilt angle of analyzer with respect to the incident optical axis. Taking Glan prism as the analyzer, we establish the first-order approximate compensation model of beam deviation based on the theory of geometric optics, acquiring the function relation between the tilt angle and wedge of Glan prism, and verify the feasibility and effectiveness of the compensation method by simulation. The study shows that the first-order approximation error of beam deviation can be compensated for by adjusting the tilt angle of Glan prism if only Glan prism is placed in a convergent beam; the tilt angle is proportional to Glan prism wedge angle, refractive index, and distance to CCD, but inversely proportional to Glan prism thickness. The results provide a theoretical basis for developing the time division imaging polarimetry with high spatial resolution and polarization accuracy.

In the letter we give new symmetries for the isospectral and non-isospectral Ablowitz-Ladik hierarchies by means of the zero curvature representations of evolution equations related to the Ablowitz-Ladik spectral problem. Lie algebras constructed by symmetries are further obtained. We also discuss the relations between the recursion operator and isospectral and non-isospectral flows. Our method can be generalized to other systems to construct symmetries for non-isospectral equations., 11 pages

M n 1⁄4 1 n; ð2aÞ ð nÞtm 1⁄4 B ðmÞ n n; ð nÞym 1⁄4 C ðmÞ n n ð2bÞ where pseudo-difference operators L and M are defined by L 1⁄4 E þ u1;n þ u2;nE 1 þ u3;nE 2 þ ; ð3Þ M 1⁄4 v0;nE 1 þ v1;n þ v2;nE þ v3;nE þ ; ð4Þ E is a shift operator; ui;n and vi;n are the functions of the variables n, t1, t2, . . ., y1, y2, . . .. The operators B ðmÞ n 1⁄4 ðLÞ 0 and CðmÞ n 1⁄4 ðMÞ 1, where ð Þ k means that part of the shift operator containing fEg ðj kÞ, and ð Þ k means that part containing fEg ðj kÞ. The 2DTL hierarchy is derived from the following four Lax equations related to eqs. (1) and (2): Ltm 1⁄4 1⁄2B ðmÞ n ; L ; ð5aÞ Lym 1⁄4 1⁄2C ðmÞ n ;L ; ð5bÞ Mtm 1⁄4 1⁄2B ðmÞ n ;M ; ð5cÞ Mym 1⁄4 1⁄2C ðmÞ n ;M : ð5dÞ Let us first consider a general formula for BðmÞ n . Set BðmÞ n 1⁄4 E m þ Xm

in which k 1⁄4 4; 5; . . . ; n. Here are some results of ðLÞþ obtained according to the formula (6): ðLÞþ 1⁄4 @; ðLÞþ 1⁄4 @ þ 2u2; ðLÞþ 1⁄4 @ þ 3u2@þ 3u3 þ 3u2;x; ðLÞþ 1⁄4 @ þ 4u2@ þ ð4u3 þ 6u2;xÞ@ þ 4u4 þ 6u3;x þ 4u2;xx þ 6u22; ðLÞþ 1⁄4 @ þ 5u2@ þ ð5u3 þ 10u2;xÞ@ þ ð5u4 þ 10u3;x þ 10u2;xx þ 10u22Þ@ þ 5u5 þ 10u4;x þ 10u3;xx þ 5u2;xxx þ 20u2u3 þ 20u2u2;x; ðLÞþ 1⁄4 @ þ 6u2@ þ ð6u3 þ 15u2;xÞ@ þ ð6u4 þ 15u3;x þ 20u2;xx þ 15u22Þ@ 2

A negative KdV-mKdV hierarchy is presented through the KdV-mKdV operator. The generalized Wronskian solution to the negative KdV-mKdV equation is obtained. Some soliton-like solutions and a complexiton solution are presented explicitly as examples.

j 1⁄4 B kj Ak j ; e Ajl 1⁄4 ðkj klÞ ðkj þ klÞ ; ð9bÞ and the sum over 1⁄4 0; 1 refers to each of the j; j 1⁄4 1; 2; ;N. Obviously, the solution of SG equation and MKdV equation can be found from (2) and (9). In addition, the solution of this equation possesses an another representation, the Wronskian form, which is essentially the same as (9). In this paper, an alternative choice of f ð1Þ of our interest is f ð1Þ 1⁄4 XN

Direct linearization method is used to solve the non-isospectral KdV equation. The corresponding singular linear integral equation and the time dependence of measure in the singular linear integral equation are proposed. Furthermore, the solutions to the non-isospectral modified KdV equation are also derived by using the singular linear integral equation of the non-isospectral KdV equation.

The bilinear form of the four-potential isospectral Ablowitz—Ladik (AL) equation is derived by the dependent variable transformation. The N-soliton solutions of the equation are obtained through the Hirota method. Moreover, the double Casoratian solution is found by means of the double Casoratian technique.

Starting from the Toda spectral problem, a mixed hierarchy of the isospectral equations and the nonisospectral equations is constructed through discrete zero curvature equations. Soliton solutions of the hierarchy are obtained by the inverse scattering transform. Meanwhile, a new kind of time evolution of scattering data for the Toda spectral problem is presented.

Bilinear forms of the non-isospectral derivative nonlinear Schrodinger equation are derived. The N-soliton solutions of this equation are obtained by Hirota's method.

The hierarchy of the nonisospectral KdV equation with self-consistent sources is derived. N-soliton solutions of the hierarchy are obtained through the inverse scattering transform. We develop new treatment of singularities appearing in the Lax pair in the process of determining time revolutions of scattering data. Our approach is general and can apply to other (1 + 1)-dimensional nonisospectral soliton hierarchy with self-consistent sources.