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The solution of the macroscopic fluctuation theory (MFT) equation can describe the optimal path of the process, and the Darboux transformation (DT) method can solve soliton solution of some integrable equations. In this paper, we obtained the exact solutions of the coupled macroscopic fluctuation theory (CMFT) equations using the DT method. By constructing a novel type of Lax pairs with i k , we derive some expressions for the 1-soliton, 2-soliton, and n -soliton solutions of the CMFT equations, including some soliton solutions, breather solutions and rational wave solutions. Based on these solutions, we consider the elastic interactions and dynamics between two solitons in CMFT equations. These results can present some novel phenomena in the optimal path of the process. [ABSTRACT FROM AUTHOR]

This paper aims to study the evolution dynamics of resonant solutions of the Davey–Stewartson II equation. The resonant solutions can depict diverse collision scenarios among periodic solitons themselves or among periodic solitons with algebraic decaying solitons. A significant finding in these particular collisions is the observation of wave structure transitions in the algebraic decaying solitons, along with notable changes in their dynamics. We show that the algebraic decaying solitons undergo a transition from states with non-localized behaviour in either space or time to states with localized behaviour in either space or time, or to states with localized behaviour in both time and space. The algebraic decaying solitons, exhibiting dual localization in both time and space, closely resemble two-dimensional localized waves in physical systems. They serve as effective tools for comprehending various nonlinear physical phenomena. • The general resonant solutions to the DSII equation. • Dynamics of the partial resonant collision of rational solitary waves with periodic solitons. • Dynamics of the complete resonant collision of rational solitary waves with periodic solitons. [ABSTRACT FROM AUTHOR]

The fifth-order coupled extended modified Korteweg–de-Vries (KdV) equation is studied. Based on seed solutions and Lax pairs, the Nth-order iterative expression of the localized wave solutions of the equation are obtained by the generalized Darboux transformation. Then, through numerical simulation, the evolution plots of the interaction of rogue waves with dark–bright solitons and the Kuznetsov–Ma breathers are derived. The results demonstrate the abundant dynamical patterns of localized waves in the fifth-order coupled systems. • Third-order localized wave solutions are studied for a fifth-order coupled extended modified KdV equation. • A new form of seed solutions is supposed. • A novel rogue wave with dark–bright solitons and the Kuznetsov–Ma breathers are derived. • Display some interesting and novel plots. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the higher-order Chen–Lee–Liu (HOCLL) equation which can describe optical pulses propagation in the medium involving high-order dispersion and quintic nonlinearity effects. Through Darboux transformation (DT), different types of solutions on the periodic and double-periodic backgrounds are constructed including periodic solutions, solitons, breathers, rogue waves and their coexistences. The modulation instability (MI) conditions and the existence region of rogue waves are analyzed and it is concluded that the region is strictly consistent with the zero frequency MI conditions. In addition, transition state mechanisms of breather-to-soliton and rogue wave-to-soliton are explored. • Solutions on periodic backgrounds are derived. • The existence region of rogue waves and modulation instability conditions are analyzed. • The transition state mechanisms are discussed. [ABSTRACT FROM AUTHOR]

This paper mainly discusses the numerical solution of Sine–Gordon (SG) equation which is widely used in engineering field. Different from the classical numerical calculation method, we choose the barycentric Lagrange interpolation collocation method (BLICM) to solve the SG equation. Firstly, the barycentric Lagrange interpolation is introduced and its differential matrix is given. Secondly, two linearized iterative schemes namely direct linearized iterative scheme and Newton linearized iterative scheme to solve SG equation are constructed, the matrix equation of the two iterative scheme is obtained. Thirdly, the Newton–Raphson iterative scheme for SG equation is also presented and the detailed solution process is given. Finally, several numerical examples with exact solution are given to show the accuracy of BLICM, the results of three iterative schemes with equidistant nodes and Chebyshev nodes are compared. Further, some examples with line and ring solitons are given to simulate by BLICM, the accurate numerical results are obtained to support our experiments. [ABSTRACT FROM AUTHOR]

We present new exotic numerical solutions and a strange periodical mixed Sasa-Satsuma and Kuznetsov-Ma waves' train. Much more, we study a nonlinear Schrödinger (NLS) equation for absorption regime in a left-handed material. The complex coefficients coming from absorption regime transform this NLS equation into a Cubic-Quintic Ginzburg–Landau (CQGL) equation. Moreover, we employ variational theory with Type I Ansatz function containing seven coordinates compared to six coordinates often used. Some interesting results are found such as one, two, three, four, five, six and seven peak solutions. Among those solutions specific ones are outlined as follows. (i) Double Kuznetsov-Ma breathers and tree structure. (ii) Conventional, random, double, triple and new forms of Sasa-Satsuma solutions. (iii) Stair-, Pyramid-, flower-shaped three and five peak solutions. (iv) First-and second-forms of seven peak solutions. Further, we also present the physical conditions and the internal excitation leading to the generation of those new exotic solutions. • The NLS equation is transform into a Cubic-Quintic Gnizburg-Landau (CQGL) equation. • We employ variational theory with Type I Ansatz function containing seven coordinates. • We find at the first time a strange periodical mixed Sasa-Satsuma and Kuznetsov-Ma waves' train. • We find some one, two, three, four, five, six and seven peak solutions. Among them emerge some exotic solitons. • We also present the physical conditions and the internal excitation leading to the generation of those exotic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the coupled Higgs equation and its multi-component generalization based on the Hirota's direct method. One and two-soliton solutions of the coupled Higgs equation are derived by the perturbation approach. We express the N -soliton solutions in the form of Pfaffians and demonstrate that the N -component coupled Higgs equation turns out to be the Pfaffian identity. One and two-soliton solutions of the multi-component coupled Higgs equation are obtained from the Pfaffians. Starting from the explicit solutions, parallel solitons, periodic and nearly periodic interactions, elastic and inelastic collisions are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we mainly focus on the Riemann–Hilbert approach solving the general n -component third-fifth-order mKdV (n -(3,5)-mKdV) equations containing the n -component mKdV equation, fifth-order mKdV equation, and their combination. Starting from the spectral analysis of the (n + 1) -order matrix Lax pair, we give the corresponding (n + 1) -order matrixed-type Riemann–Hilbert problem. By solving the Riemann–Hilbert problem, N -soliton solutions of the n -(3,5)-mKdV equations can be found. Particularly, for the case of reflectionless and some types of spectral parameters of the Lax pair, we analyze the interactions of some kinds of solutions of the 3-component mKdV equation, fifth-order mKdV equation, and third-fifth-order mKdV equations, including breather solutions, W -shaped solutions, anti-bright and bright solutions. Some elastic collisions between them are also presented. In addition, the multi-pole solutions are obtained for the n -(3,5)-mKdV equations through the L'Hospital's rule. These results will be useful to further understand the related wave phenomena in the multi-component physical systems. • The (n + 1) -order matrixed-type Riemann–Hilbert problem is presented. • N -soliton solutions are found for the n -(3,5)-mKdV equations. • The multi-pole solutions are obtained for the n -(3,5)-mKdV equations through the L'Hospital's rule. [ABSTRACT FROM AUTHOR]

In this paper, a nonlocal Lakshmanan–Porsezian–Daniel equation is investigated with the help of the binary Darboux transformation method and asymptotic analysis. Nonlocality of that equation has been reflected in that the solutions of that equation at the location ς depend on both the local solution at ς and the nonlocal solution at − ς , where ς is the retarded time coordinate. We derive the formulas of the N th-order solutions through the obtained binary Darboux transformation, where N is a positive integer. Under certain conditions, the first-order periodic waves and solitons are obtained, e.g., degenerate solitons, dark–dark solitons, bright–bright solitons and dark–bright solitons. Interactions between/among the dark solitons, bright solitons and periodic wave are discussed and graphically illustrated. We discuss the modulation instability of that equation. [ABSTRACT FROM AUTHOR]

In this paper, we inspect some novel localized waves solutions for the n -coupled Kundu–Eckhaus (KE)-equations (n ≥ 3). The Lax pair for the 3-coupled KE equations is constructed and generalized to the n -components KE equations. In this way, for any n ∈ N ∗ , we determine the (n + 1) × (n + 1) Darboux matrix and the explicit expressions of both one-soliton solutions and two-soliton solutions. Furthermore, we apply the Darboux-dressing transformation to the n -coupled KE equations with non-vanishing background. Based on this technique, we derive the rogue waves and solutions for the 3-coupled KE equations. An appropriate choice of parameters allows us to appreciate the dynamics of soliton solutions for high values of n : n = 3 , 35 , 50 , and the dynamics of the rogue waves interacting with breathers for n = 3. • One and two-soliton solutions for any system of n-coupled KE equations are given. • The Darboux-dressing transformation of n-coupled KE equations is provided. • Novel Rogue waves solutions and new profiles are obtained for n=3. [ABSTRACT FROM AUTHOR]

In this paper, we study the vector multi-pole (MP) solutions of the r -coupled Hirota equation (RCHE). First of all, based on the Darboux transformation and the N -soliton solution, we derive the explicit formulas of the arbitrary-order vector MP solutions. Then, through the balance between exponential and algebraic terms, we give all asymptotic solitons in the vector double- and triple-pole solutions via an asymptotic analysis method. Furthermore, we analyze the soliton interactions in vector MP solutions. By comparing with the scalar Hirota equation, we find that the RCHE has richer collision properties: the position shift of each component soliton grows logarithmically with r , the asymptotic solitons have the same shape in each component, and their amplitudes are proportional among the components. • Derive the explicit formulas of the arbitrary-order vector MP solutions. • Analyze the soliton interactions in vector MP solutions. • Find that the r-coupled Hirota equation has richer collision properties. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a general decomposition for solutions obtained through the Hirota substitution method from linear combinations of exponents. In particular, we study the Wronskian solutions for the good Boussinesq (gB) equation. The asymptotical analysis is carried in terms of multi-linear functions. We show that a multi-soliton solution for the good Boussinesq equation, which involves interaction of two resonant solitons, always develops singularity. • General decomposition for solutions obtained through the Hirota substitution method. • Wronskian solutions for the good Boussinesq (gB) equation. • The asymptotical analysis is carried in terms of multi-linear functions. • Two resonant soliton solution for gB always develops singularity. [ABSTRACT FROM AUTHOR]

This study derives solitons solutions of the reverse-time nonlocal Davey–Stewartson III equation by the Kadomtsev–Petviashvili hierarchy reduction method and Hirota's bilinear method. The solutions are expressed as N × N Gram-type determinants with different parametric reduction conditions. N -soliton and line breather solutions on both constant and periodic backgrounds are derived. The dynamics of these solutions are discussed. All possible configurations of these solutions are illustrated for 1 ≤ N ≤ 4. Both intersecting and parallel solitons are presented. In particular, the elastic and inelastic collisions of the two parallel-soliton solutions are determined. In the inelastic case, the amplitudes of the solitons change after collision. Moreover, the parametric conditions for determining the inelastic collisions are derived, and all possible types of inelastic behaviors are obtained and displayed. • N -soliton solutions on constant and periodic backgrounds are obtained. • Interesting inelastic collisions between solitons are discussed in detail. • Breather solutions on constant and periodic backgrounds are obtained. [ABSTRACT FROM AUTHOR]