Narrow Results

In the shallow-water studies, the Whitham–Broer–Kaup (WBK) system can be used to describe the propagation of the long waves. In this paper, based on the Lax pair of the WBK system, we derive the gauge transformation from the WBK system to the Ablowitz–Kaup–Newell–Segur (AKNS) system with the help of symbolic computation. Applying the Darboux transformation of the AKNS system, we obtain some soliton solutions of the WBK system. Those results might be useful in the investigations on the propagation of solitons in such situation as shallow water.

In this paper, a (3+1)-dimensional generalized variable-coefficients Kadomtsev–Petviashvili (gvcKP) equation is proposed, which describes many nonlinear phenomena in fluid dynamics and plasma physics. By a very natural way, the integrable constraint conditions on the variable coefficients are presented to investigate the integrabilities of the gvcKP equation. Based on the generalized Bell's polynomials, we succinctly obtain its bilinear representations, bilinear Bäcklund transformation and Lax pair, respectively. Furthermore, by virtue of the binary Bell polynomial form, the infinite conservation laws of the equation are found with explicit recursion formulas as well by using its Lax equations via algebraic and differential manipulation. In addition, by using the Hirota bilinear method, its N-soliton solutions are also obtained.

The transition phenomenon of few-cycle-pulse optical solitons from a pure modified Korteweg–de Vries (mKdV) to a pure sine-Gordon regime can be described by the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients. Based on the Bell polynomials, Hirota method and symbolic computation, bilinear forms and soliton solutions for this equation are obtained. Bäcklund transformations (BTs) in both the binary Bell polynomial and bilinear forms are obtained. By virtue of the BTs and Ablowitz–Kaup–Newell–Segur system, Lax pair and infinitely many conservation laws for this equation are derived as well.

Under investigation in this paper is a (2+1)-dimensional B-type Kadomtsev–Petviashvili equation for the shallow water wave in a fluid or electrostatic wave potential in a plasma. Bilinear form, Bäcklund transformation and Lax pair are derived based on the binary Bell polynomials. Multi-soliton solutions are constructed via the Hirota’s method. Propagation and interaction of the solitons are illustrated graphically: (i) Through the asymptotic analysis, elastic and inelastic interactions between the two solitons are discussed analytically and graphically, respectively. The elastic interaction, amplitudes, velocities and shapes of the two solitons remain unchanged except for a phase shift. However, in the area of the inelastic interaction, amplitudes of the two solitons have a linear superposition. (ii) Elastic interactions among the three solitons indicate that the properties of the elastic interactions among the three solitons are similar to those between the two solitons. Moreover, oblique and overtaking interactions between the two solitons are displayed. Oblique interactions among the three solitons and interactions among the two parallel solitons and a single one are presented as well. (iii) Inelastic–elastic interactions imply that the interaction between the inelastic region and another one is elastic.

A new Lax pair is introduced for a perturbed Kaup–Newell equation and used to construct two series of conservation laws through a Riccati equation that a ratio of eigenfunctions satisfies. Both series of conservation laws are defined recursively, and the first two in each series are presented explicitly.

Under investigation in this paper is a generalized variable-coefficient Boussinesq system, which describes the propagation of the shallow water waves in the two-layered fluid flow. Bilinear forms, Bäcklund transformation and Lax pair are derived by virtue of the Bell polynomials. Hirota method is applied to construct the one- and two-soliton solutions. Propagation and interaction of the solitons are illustrated graphically: kink- and bell-shape solitons are obtained; shapes of the solitons are affected by the variable coefficients ${\alpha }_1$, ${\alpha }_3$ and ${\alpha }_4$ during the propagation, kink- and anti-bell-shape solitons are obtained when ${\alpha }_3>0$, anti-kink- and bell-shape solitons are obtained when ${\alpha }_3<0$; Head-on interaction between the two bidirectional solitons, overtaking interaction between the two unidirectional solitons are presented; interactions between the two solitons are elastic.

We point out that the Darboux transformation in the paper “Analytic solutions and Darboux transformation to a new Hamiltonian lattice hierarchy” [Mod. Phys. Lett. B30 (2016) 1650100] is not right, and present a correct Darboux transformation.

Under investigation in this paper is a (2+1)-dimensional generalized variable-coefficient shallow water wave equation which can be reduced to several integrable equations, such as the Korteweg–de Vries (KdV) equation and the Calogero–Bogoyavlenskii–Schiff (CBS) equation. Bilinear forms, Bäcklund transformation, Lax pair and infinite conservation laws are derived based on the binary Bell polynomials. N-soliton solutions are constructed via the Hirota method. Propagation and interaction of the solitons are illustrated graphically: (i) variable coefficients affect the shape of the N-soliton interaction in the scaled space and time coordinates; (ii) positions of the solitons depend on the sign of wave numbers after each interaction; (iii) interaction of the solitons is elastic, i.e. the amplitude, velocity and shape of each soliton remain invariant after each interaction except for a phase shift.

In this paper, an eighth-order nonlinear Schrödinger equation is investigated in an optical fiber, which can be used to describe the propagation of ultrashort nonlinear pulses. Lax pair and infinitely-many conservation laws are derived to verify the integrability of this equation. Via the Darboux transformation and generalized Darboux transformation, the analytic breather and rogue wave solutions are obtained. Influence of the coefficients of operators in this equation, which represent different order nonlinearity, and the spectral parameter on the propagation and interaction of the breathers and rogue waves is also discussed. We find that (i) the periodic of the breathers decreases as the augment of the spectral parameter; (ii) the coefficients of operators change the compressibility and periodic of the breathers, and can affect the interaction range and temporal–spatial distribution of the rogue waves.

For describing the propagation of ultrashort pulses in the birefringent or two-mode fiber, we consider the coupled nonlinear Schrödinger equations with higher-order effects. According to the Ablowitz–Kaup–Newell–Segur system, 3$\times$3 Lax pair for such equations is derived. Based on the Lax pair, we construct the conservation laws and Darboux transformation (DT). One- and two-soliton solutions are obtained via the DT, and we graphically present the one soliton and bound-state two solitons.

In this paper, a vector Ramani equation is proposed by using the bilinear approach. With the help of the bilinear exchange formulae, bilinear Bäcklund transformation and the corresponding Lax pair for the vector Ramani equation are derived. Besides, multi-soliton solution expressed by pfaffian is given and proved by pfaffian techniques.

Under investigation with symbolic computation in this paper, is a variable-coefficient Sasa–Satsuma equation (SSE) which can describe the ultra short pulses in optical fiber communications and propagation of deep ocean waves. By virtue of the extended Ablowitz–Kaup–Newell–Segur system, Lax pair for the model is directly constructed. Based on the obtained Lax pair, an auto-Bäcklund transformation is provided, then the explicit one-soliton solution is obtained. Meanwhile, an infinite number of conservation laws in explicit recursion forms are derived to indicate its integrability in the Liouville sense. Furthermore, exact explicit rogue wave (RW) solution is presented by use of a Darboux transformation. In addition to the double-peak structure and an analog of the Peregrine soliton, the RW can exhibit graphically an intriguing twisted rogue-wave (TRW) pair that involve four well-defined zero-amplitude points.

A Toda lattice hierarchy is studied by introducing a new spectral problem which is a discrete counterpart of the generalized Kaup–Newell spectral problem. Based on the Lenard recursion equation, Lax pair of the hierarchy is given. Further, the discrete spectral problem is nonlinearized into an integrable symplectic map. As a result, an algebraic–geometric solution in Riemann theta function of the hierarchy is obtained. Besides, two equations, the Volterra lattice and a (2+1)-dimensional Burgers equation with a discrete variable, yielded from the hierarchy are also solved.

In this work, we study the modified Volterra lattice. Applying the gauge transformation of the associated $2\times 2$ matrix spectral problems, we establish the N-fold Darboux transformation (DT), and then construct a few explicit solutions in terms of determinants upon using the obtained DT. Moreover, all the results are illustrated by the graphs of the solitonic evolution profiles of the aforementioned solutions. Finally, infinitely many conservation laws for the modified Volterra lattice are proposed. The obtained results of this research might be applied to the research on nonlinear phenomena in physics or engineering areas.

In this paper, we study a Kundu–Eckhaus equation with variable coefficients, which describes the ultra-short optical pulses in an inhomogeneous optical fiber. We construct the Lax pair under certain variable-coefficient constraints. With the gauge transformation, one/N-fold binary Darboux transformations and limit forms of the one-fold binary Darboux transformation are obtained. Based on such transformations, one/N-dark (N = 2,3, $\dots$) soliton solutions under those constraints are derived. Linear, periodic and parabolic dark solitons are presented, and numerical simulations are used to investigate the influence of the group velocity dispersion on the structures of the one-dark solitons. Based on the two-dark soliton solutions under certain variable-coefficient constraints, we also discuss the influence of the group velocity dispersion on the structures of the two-dark solitons. Head-on and overtaking collisions between the two linear, parabolic and cubic-type dark solitons are presented.

Hirota equation is a modified nonlinear Schrödinger (NLS) equation, which takes into account higher order dispersion and delay correction of cubic nonlinearity. The propagation of the waves in the ocean is described, and the optical fiber can be regarded as a more accurate approximation than the NLS equation. Using the algebraic reductions from the Lie algebra $gl(n,ℂ)$ to its commutative subalgebra $Z_n$, we construct the general $Z_n$-Hirota systems. Considering the potential applications of two-mode nonlinear waves in nonlinear optical fibers, including its Lax pairs, we use the algebraic reductions of the Lie algebra $gl(2,ℂ)$ to its commutative subalgebra $Z_2=ℂ[\mathrm{\Gamma }]/({\mathrm{\Gamma }}^2)$. Then, we construct Darboux transformation of the strongly coupled Hirota equation, which implies the new solutions of $(q^{[1]},r^{[1]})$ generated from the known solution $(q,r)$. The new solutions $(q^{[1]},r^{[1]})$ furnish soliton solutions and breather solutions of the strongly coupled Hirota equation. Furthermore, using Taylor series expansion of the breather solutions, the rogue waves of the strongly coupled Hirota equation can be given demonstrably. It is obvious that different images can be obtained by choosing different parameters.

A new coupled Burgers equation and a new coupled KdV equation which are associated with $3\times 3$ matrix spectial problem are investigated for complete integrability and covariant property. For integrability, Lax pair and conservation laws of the two new coupled equations with four potentials are established. For covariant property, Darboux transformation (DT) is used to construct explicit solutions of the two new coupled equations.

In this paper, the Riemann–Hilbert approach is applied to investigate a higher-order Chen–Lee–Liu equation with third-order dispersion and quintic nonlinearity terms. Based on the analytical, symmetric and asymptotic properties of eigenfunctions, a generalized Riemann–Hilbert problem associated with Chen–Lee–Liu equation with nonzero boundary conditions is constructed. Further, the $N$-soliton solution is found by solving the generalized Riemann–Hilbert problem. As an illustrative example, two kinds of one-soliton solutions with different forms of parameters are obtained.

In this paper, we deduce the (2+1)-dimensional Schwarz–Korteweg–de Vries equation from two (1+1)-dimensional equations. Based on the resulting Lax pairs, we present its $N$-fold Darboux transformation. From a trivial solution, we get $2N$-soliton and $(2N+1)$-soliton solutions of the (2+1)-dimensional Schwarz–Korteweg–de Vries equation. The asymptotic analyses of the $2$-soliton, $3$-soliton and $4$-soliton solutions are presented theoretically and graphically.

Binary Bell polynomial approach is applied to study the coupled Ramani equation, which is the generalization of the Ramani equation. Based on the concept of scale invariance, the coupled Ramani equation is written in terms of binary Bell polynomials of two dimensionless field variables, which leads to the bilinear coupled Ramani equation directly. As a consequence, the bilinear Bäcklund transformation, Lax pair and conservation laws are systematically constructed by virtue of binary Bell polynomials.