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Darboux transformation for the supersymmetric nonisospectral KdV equation is investigated. Based on the Lax pair, we successively construct the one-fold, two-fold and three-fold Darboux tansformations for the supersymmetric nonisospectral KdV equation. Moreover, we present the $n$-fold Darboux transformation in the form of superdeterminant.

In this paper, we study a coupled system of the nonlinear Schrödinger (NLS) equation and the Maxwell–Bloch (MB) equation with nonzero boundary conditions by Riemann–Hilbert (RH) method. We obtain the formulae of the simple-pole and the multi-pole solutions via a matrix Riemann–Hilbert problem (RHP). The explicit form of the soliton solutions for the NLS-MB equations is obtained. The soliton interaction is also given. Furthermore, we show that the multi-pole solutions can be viewed as some proper limits of the soliton solutions with simple poles, and the multi-pole solutions constitute a novel analytical viewpoint in nonlinear complex phenomena. The advantage of this way is that it avoids solving the complex symmetric relations and repeatedly solving residue conditions.

Burgers-type equations are considered as the models of certain phenomena in plasma astrophysics, ocean dynamics, atmospheric science and so on. In this paper, a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves is studied. Based on the Painlevé-Bäcklund equations, one auto-Bäcklund transformation and two hetero-Bäcklund transformations are derived. Motivated by the Burgers hierarchy, a Lax pair is given. Via two hetero-Bäcklund transformations with different constant seed solutions, we find some multiple-kink solutions, complex periodic solutions, hybrid solutions composed of the lump, periodic and multiple kink waves. Then we discuss the influence of the coefficients of the above equation on such solutions. Via the auto-Bäcklund transformation with the nontrivial seed solutions, we obtain certain lump-type solutions, kink-type solutions and recurrence relation of the above equation.

For a nonlinear Schrödinger–Hirota equation with the spatio-temporal dispersion and Kerr law nonlinearity in nonlinear optics, we derive a Lax pair, a Darboux transformation and two families of the periodic-wave solutions via the Jacobian elliptic functions dn and cn. We construct the linearly-independent and non-periodic solutions of that Lax pair, and substitute those solutions into the Darboux transformation to get the rogue-periodic-wave solutions. When the third-order dispersion or group velocity dispersion (GVD) or inter-modal dispersion (IMD) increases, the maximum amplitude of the rogue-periodic wave remains unchanged. From the rogue-dn-periodic-wave solutions, when the GVD decreases, the minimum amplitude of the rogue-dn-periodic wave decreases. When the third-order dispersion decreases, the minimum amplitude of the rogue-dn-periodic wave rises. Decrease of the IMD causes the period of the rogue-dn-periodic wave to decrease. From the rogue-cn-periodic-wave solutions, when the GVD increases, the minimum amplitude of the rogue-cn-periodic wave decreases. Increase of the third-order dispersion or IMD leads to the decrease of the period.

To give rigorous mathematical proofs of chaotic behaviors in a given system, it is necessary to identify the homoclinic structures in the system. In this tutorial review, methods for constructing explicit solutions for nonlinear partial differential equations are presented, with more emphasis placed on those utilizing complete integrability associated with soliton equations. As an extended application, homoclinic orbits to spatial uniform plane waves of coupled modified nonlinear Schrödinger equations are obtained via the dressing method. During the procedure, it is necessary to introduce the Lax pair for these coupled equations, as well as its Floquet spectral analysis and corresponding Bloch functions.

One of the authors recently introduced the concept of conjugate Hamiltonian systems: the solution of the equation h = H(p, q, t), where H is a given Hamiltonian containing t explicitly, yields the function t = T(p, q, h), which defines a new Hamiltonian system with Hamiltonian T and independent variable h. By employing this construction and by using the fact that the classical Painlevé equations are Hamiltonian systems, it is straightforward to associate with each Painlevé equation two new integrable ODEs. Here, we investigate the conjugate Painlevé II equations. In particular, for these novel integrable ODEs, we present a Lax pair formulation, as well as a class of implicit solutions. We also construct conjugate equations associated with Painlevé I and Painlevé IV equations.

We have considered soliton pulse compression in a dispersion decreasing ideal elliptical birefringent fiber with phase modulation. We have shown that there is exact balancing between the effective gain and the effective phase modulation and as a result arrive at the fundamental soliton solution. For the various dispersion decreasing profiles we have considered, we have shown that the hyperbolic profile is the best suitable choice for the dispersion decreasing profile as it provides fairly good compression even after compensating for the fiber loss.

The orthogonal polynomials on the unit circle associated with the unitary matrix model have various interesting properties, and have been studied in connection with different applications, such as double scaling, Riemann–Hilbert problems and integrable Fredholm operators. In this paper, we study the orthogonal polynomials on the unit circle with the weight function . We use the orthogonality of the polynomials to show that the orthogonal polynomials on the unit circle satisfy the linear problems associated with the discrete PainlevéII hierarchy, alternate discrete PainlevéII hierarchy and the discrete MKdV hierarchy. Thus the isomonodromy deformation method can be used to study the unitary matrix model. Typically, we focus on the orthogonal polynomials with the weight function exp{s(z+z^-1)}. The recursion formula z(p_n+v_np_n-1)=p_n+1+u_np_n (derived from Szegö's equation) for the orthogonal polynomials p_n(z,s) is proved to be equivalent to the discrete AKNS-ZS system which is the base equation in the linear problem for the discrete equations. The variable x_n=p_n(0,s) satisfies the discrete PainlevéII equation and the discrete MKdV equation; u_n, v_n satisfy the alternate discrete PainlevéII equation; 1/u_n-1=-x_n-1/x_n satisfies the PainlevéIII equation; and satisfies the PainlevéV equation. Also, we discuss the continuum limits of the discrete equations by using the double scaling method. Further, we present a procedure for finding the linear equations for the orthogonal polynomials and the consistency conditions of the linear equations by using the orthogonality of the polynomials.

It is shown how a system of differential forms can reproduce the complete set of differential equations generated by an SO(m) matrix Lax pair. By selecting the elements in the given matrices appropriately, examples of integrable nonlinear equations can be produced. The SO(3) case is discussed in detail, then extended to an m - 1 dimensional manifold immersed in Euclidean space.

According to the polynomial recursion formalism, the modified Jaulent–Miodek hierarchy is derived in a standard way. The first two nontrivial members in the modified Jaulent–Miodek hierarchy are listed correspondingly. Based on the squared eigenfunctions, an algebraic curve ${\kappa }_n$ and a Riemann surface $S$ with arithmetic genus $n$ are introduced, then the Dubrovin-type equations are obtained naturally. With the help of the conservation laws, the Baker–Akhiezer functions are defined. Finally, the asymptotic properties of the Baker–Akhiezer functions are analyzed, from which the algebro-geometric solutions of the modified Jaulent–Miodek hierarchy are constructed in term of the Riemann theta function.

A higher-order spectral problem is introduced, based on a special Lie subalgebra of the general linear algebra. An associated matrix Liouville integrable hierarchy, each of which consists of four submatrix equations, is generated by means of the zero curvature formulation. The corresponding Hamiltonian structure is established by the trace variational identity and two integrable reductions, of which one is real and the other is complex, are constructed under similarity transformations.

In the recent paper by one of the authors (MBS) and A. A. Malykh on the classification of second-order PDEs with four independent variables that possess partner symmetries [1], mixed heavenly equation and Husain equation appear as closely related canonical equations admitting partner symmetries. Here for the mixed heavenly equation and Husain equation, formulated in a two-component form, we present recursion operators, Lax pairs of Olver–Ibragimov–Shabat type and discover their Lagrangians, symplectic and bi-Hamiltonian structure. We obtain all point and second-order symmetries, integrals and bi-Hamiltonian representations of these systems and their symmetry flows together with infinite hierarchies of nonlocal higher symmetries.

The goal of this paper is to rederive the connection between the Painlevé 5 integrable system and the universal eigenvalues correlation functions of double-scaled Hermitian matrix models, through the topological recursion method. More specifically we prove, to all orders, that the WKB asymptotic expansions of the τ-function as well as of determinantal formulas arising from the Painlevé 5 Lax pair are identical to the large N double scaling asymptotic expansions of the partition function and correlation functions of any Hermitian matrix model around a regular point in the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to O(N^-5).