Sasa Satsuma Equation Research Articles

Riemann–Hilbert Method for the Three-Component Sasa–Satsuma Equation and Its N-Soliton Solutions

The three-component Sasa–Satsuma equation associated with the 7 × 7 matrix spectral problem is studied by using the Riemann–Hilbert method. The spectral analysis of the Lax pair is performed for the three-component Sasa–Satsuma equation, from which a Riemann–Hilbert problem is formulated. As applications, N-soliton solutions for the three-component Sasa–Satsuma equation are obtained by solving the Riemann–Hilbert problems corresponding the reflectionless case. Furthermore, a compact form of N-soliton solutions formula is given explicitly, which is the ratio of (2N + 1) × (2N + 1)-determinant and (2N × 2N)-determinant.

Long-time asymptotics for the generalized Sasa-Satsuma equation

In this paper, we study the long-time asymptotic behavior of the solution of the Cauchy problem for the generalized Sasa-Satsuma equation. Starting with the 3 × 3 Lax pair related to the generalized Sasa-Satsuma equation, we construct a Rieman-Hilbert problem, by which the solution of the generalized Sasa-Satsuma equation is converted into the solution of the corresponding RiemanHilbert problem. Using the nonlinear steepest decent method for the Riemann-Hilbert problem, we obtain the leading-order asymptotics of the solution of the Cauchy problem for the generalized SasaSatsuma equation through several transformations of the Riemann-Hilbert problem and with the aid of the parabolic cylinder function.

An integrable reverse space-time nonlocal Sasa-Satsuma equation

In this paper, we introduce an integrable reverse space-time nonlocal Sasa-Satsuma equation. The Darboux transformation and soliton solutions for this nonlocal integrable equation are constructed.

Asymptotics for the Sasa–Satsuma equation in terms of a modified Painlevé II transcendent

We consider the initial-value problem for the Sasa–Satsuma equation on the line with decaying initial data. Using a Riemann–Hilbert formulation and steepest descent arguments, we compute the long-time asymptotics of the solution in the sector |x|≤Mt1/3, M constant. It turns out that the asymptotics can be expressed in terms of the solution of a modified Painlevé II equation. Whereas the standard Painlevé II equation is related to a 2×2 matrix Riemann–Hilbert problem, this modified Painlevé II equation is related to a 3×3 matrix Riemann–Hilbert problem.

Intricate dynamics of rogue waves governed by the Sasa–Satsuma equation

We present the second order rogue wave solutions of the Sasa–Satsuma evolution equation that have significantly more complicated structure than in the case of the nonlinear Schrödinger equation. This equation takes into account the third order dispersion, self-frequency shift and self-steepening effects, thus making it closer to realistic physical applications. These effects are important both for water waves and pulses in optical fibres. Although the Sasa–Satsuma equation is integrable, the degree of complexity of these new solutions has prevented them being found previously. Here, we present in explicit form the second-order rogue wave solutions of the Sasa–Satsuma equation for three free parameter values of the family of rogue wave solutions. Special choices of the free parameters allow us to present these solutions in the simplest possible way and to illustrate them graphically.

An initial-boundary value problem for the generalized Sasa-Satsuma equation on the half-line

An initial-boundary value problem for the generalized Sasa-Satsuma equation on the half-line

Optical solitons to Sasa–Satsuma model with trial equation approach

Optical soliton pulses or molecules with the Sasa–Satsuma equation have been worked by trial equation methodology in this paper. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton, bright optical soliton in additionally singular optical soliton.

Optical solitons to Sasa-Satsuma model in birefringent fibers with modified simple equation approach

Optical soliton pulses or molecules with the Sasa-Satsuma equation in birefringent fibers have been worked by modified simple equation methodology in this paper. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton, bright optical soliton in additionally singular optical soliton.

Optical solitons to Sasa-Satsuma model with modified simple equation approach

Optical soliton pulses or molecules with the Sasa-Satsuma equation have been worked by modified simple equation methodology in this paper. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton, bright optical soliton in additionally singular optical soliton.

Optical solitons to Sasa–Satsuma model in birefringent fibers with trial equation approach

Optical soliton pulses or molecules with the Sasa–Satsuma equation in birefringent fibers have been worked by trial equation methodology in this paper. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton, bright optical soliton in additionally singular optical soliton.

L2-Sobolev space bijectivity of the inverse scattering of a 3 × 3 AKNS system

We study the -Sobolev space bijectivity of the direct and inverse scattering of the AKNS system associated to the Manakov system and Sasa–Satsuma equation. We establish the bijectivity on the weighted Sobolev space for .

Long-time asymptotics for the Sasa–Satsuma equation via nonlinear steepest descent method

We formulate a 3 × 3 Riemann–Hilbert problem to solve the Cauchy problem for the Sasa–Satsuma equation on the line, which allows us to give a representation for the solution of the Sasa–Satsuma equation. We then apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Sasa–Satsuma equation.

Generalised Sasa–Satsuma Equation: Densities Approach to New Infinite Hierarchy of Integrable Evolution Equations

Abstract We derive the new infinite Sasa–Satsuma hierarchy of evolution equations using an invariant densities approach. Being significantly simpler than the Lax-pair technique, this approach does not involve ponderous 3 × 3 matrices. Moreover, it allows us to explicitly obtain operators of many orders involved in the time evolution of the Sasa–Satsuma hierarchy functionals. All these operators are parts of a generalised Sasa–Satsuma equation of infinitely high order. They enter this equation with independent arbitrary real coefficients that govern the evolution pattern of this multiparameter dynamical system.

High-order soliton matrices for Sasa–Satsuma equation via local Riemann–Hilbert problem

A study of high-order soliton matrices for Sasa–Satsuma equation in the framework of the Riemann–Hilbert problem approach is presented. Through a standard dressing procedure, soliton matrices for simple zeros and elementary high-order zeros in the Riemann–Hilbert problem for Sasa–Satsuma equation are constructed, respectively. It is noted that pairs of zeros are simultaneously tackled in the situation of the high-order zeros, which is different from other NLS-type equation. Furthermore, the generalized Darboux transformation for Sasa–Satsuma equation is also presented. Moreover, collision dynamics along with the asymptotic behavior for the two-solitons are analyzed, and long time asymptotic estimations for the high-order one-soliton are concretely calculated. In this case, two double-humped solitons with nearly equal velocities and amplitudes can be observed.

The Deift–Zhou steepest descent method to long-time asymptotics for the Sasa–Satsuma equation

The initial value problem for the Sasa–Satsuma equation is transformed to a 3×3 matrix Riemann–Hilbert problem with the help of the corresponding Lax pair. Two distinct factorizations of the jump matrix and a decomposition of the vector-valued function ρ(k) are given, from which the long-time asymptotics for the Sasa–Satsuma equation with decaying initial data is obtained by using the nonlinear steepest descent method.

The initial-boundary value problem for the Sasa-Satsuma equation on a finite interval via the Fokas method

In this paper, the initial-boundary value problem for the Sasa-Satsuma equation on the finite interval is studied by using the Fokas method (or called unified transform method). We show that the solution of this problem can be expressed in terms of the solution of a 3 × 3 Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of the three matrix-value spectral functions s(k), S(k), and SL(k), which in turn are defined in terms of the initial values u0(x), boundary values at x = 0, i.e, {g0(t), g1(t), g2(t)}, and boundary values at x = L, i.e, {f0(t), f1(t), f2(t)}, respectively. However, for a well-posed problem, only part of the boundary values can be prescribed, and the remaining boundary data cannot be independently specified but are determined by the so-called global relation. As the Lax pair is 3 × 3 and the derivative order of the Sasa-Satsuma equation is three, the analysis of the global relation is more complicated. We show that if we know the initial and partial boundary values {g0(t), f0(t), f1(t)}, then we can characterize the unknown boundary values {g1(t), g2(t), f2(t)} via a Gelfand-Levitan-Marchenko representation of the eigenfunctions {Φij(t, k), ϕij(t, k)} though analyzing the global relation.

Dark-bright solitons and semirational rogue waves for the coupled Sasa-Satsuma equations.

In this paper, we investigate the coupled Sasa-Satsuma equations, which describe the simultaneous propagation of two ultrashort pulses in the birefringent or two-mode fiber with the third-order dispersion, self-steepening, and stimulated Raman scattering effects. Darboux-dressing transformation is applied to obtain the dark-bright soliton and semirational rogue-wave solutions. Dark-bright one solitons with the single-hump, double-hump, and even breather-like structures are presented. Interactions between the double-peak breather and different kinds of dark-bright solitons are studied. We show that the double-peak (or single-peak) rogue wave can coexist and interact with different kinds of dark-bright solitons. Coexistence of the solitons with different velocities and rogue waves is also found. Numerical stabilities of the dark-bright solitons and semirational rogue waves are exhibited. It is expected that those localized wave phenomena can be experimentally observed and have potential applications.

Sasa-Satsuma hierarchy of integrable evolution equations.

We present the infinite hierarchy of Sasa-Satsuma evolution equations. The corresponding Lax pairs are given, thus proving its integrability. The lowest order member of this hierarchy is the nonlinear Schrödinger equation, while the next one is the Sasa-Satsuma equation that includes third-order terms. Up to sixth-order terms of the hierarchy are given in explicit form, while the provided recurrence relation allows one to explicitly write all higher-order terms. The whole hierarchy can be combined into a single general equation. Each term in this equation contains a real independent coefficient that provides the possibility of adapting the equation to practical needs. A few examples of exact solutions of this general equation with an infinite number of terms are also given explicitly.

A super Sasa–Satsuma hierarchy and bi-Hamiltonian structures

A super Sasa–Satsuma hierarchy associated with a 3 × 3 matrix spectral problem is proposed with the aid of the zero-curvature equation and Lenard recursion equations. A typical member in the hierarchy is the super Sasa–Satsuma equation. The super bi-Hamiltonian structures of the super Sasa–Satsuma hierarchy are constructed by utilizing the super trace identity. The infinite conservation laws of the super Sasa–Satsuma equation are presented by resorting to the spectral parameter expansion.

Vector bright soliton interactions of the coupled Sasa–Satsuma equations in the birefringent or two-mode fiber

Investigated in this paper are the vector bright soliton interactions of the coupled Sasa–Satsuma equations, which describe the simultaneous propagation of two ultrashort pulses in the birefringent or two-mode fiber. Lax pair and the corresponding Darboux-dressing transformation are constructed. Performing the iterative formula, analytic bright one-, two- and even N-soliton solutions can be obtained. Through classifying different parameter conditions, we present six cases of the interactions between the vector two solitons via the graphic analysis. During the interaction between the two solitons, in addition to the energy exchanges, the singular-I soliton always keeps its sech-type profile, while the singular-II and nonsingular solitons may be converted to each other.