The main purpos e of this chapter is to present a direct and systematic way of finding exact solutions and Backlund transformations of a certain class of nonlinear evolution equations. The nonlinear evolution equations are transformed, by changing the dependent variable(s), into bilinear differential equations of the following special form 
 
$$ F\left( {\frac{\partial }{{\partial t}} - \frac{\partial }{{\partial {t^1}}},\frac{\partial }{{\partial x}} - \frac{\partial }{{\partial {x^1}}}} \right)f(t,x)f({t^1},{x^1}){|_{t = {t^1},x = {x^1}}} = 0 $$ 
 
, which we solve exactly using a kind of perturbational approach.

Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告)

A (2+1)-dimensional generalized breaking soliton system is under investigation, which can be used to describe the interaction between Riemann wave and long wave along with two space directions in nonlinear media. Rogue wave solutions, two types of mixed soliton and rogue wave solutions are constructed through appropriately choosing the functions in the bilinear forms. The rogue wave structures and the interaction characteristics between the soliton and rogue wave are studied in details. Novel hidden rogue wave and hidden soliton, which are invisible but work, are observed during the interactions.

Interactions between soliton and rogue wave for a (2+1)-dimensional generalized breaking soliton system: Hidden rogue wave and hidden soliton

A new variant of the $$(2+1)$$
 -dimensional [
 $$(2+1)d$$
 ] Boussinesq equation was recently introduced by Zhu [Line soliton and rational solutions to (2+1)-dimensional Boussinesq equation by Dbar problem, 2017. arXiv:1704.02779v2
 ; see eq. (3)]. First, we derive in this paper the one-soliton solutions of both bright and dark types for the extended $$(2+1)d$$
 Boussinesq equation by using the traveling wave method. Second, N-soliton, breather, and rational solutions are obtained by using the Hirota bilinear method and the long-wave limit. Nonsingular rational solutions of two types were obtained analytically, namely (i) rogue wave solutions having the form of W-shaped lines waves and (ii) lump-type solutions. Two generic types of semi-rational solutions were also put forward. The obtained semi-rational solutions are as follows: (iii) a hybrid of a first-order lump and a bright one-soliton solution and (iv) a hybrid of a first-order lump and a first-order breather.

Families of exact solutions of a new extended $$\varvec{(2+1)}$$ ( 2 + 1 ) -dimensional Boussinesq equation

Lump-type solutions and interaction solutions for the (2+1)-dimensional generalized fifth-order KdV equation

Rogue wave solutions for a higher-order nonlinear Schrödinger equation in an optical fiber

Under investigation in this paper is a sixth-order variable-coefficient nonlinear Schrodinger equation in an ocean or optical fiber. Through the Darboux transformation (DT) and generalized DT, we obtain the multi-soliton solutions, breathers and rogue waves. Choosing different values of $\alpha$
 (x), $\beta$
 (x), $\gamma$
 (x) and $\delta$
 (x), which are the coefficients of the third-, fourth-, fifth- and sixth-order dispersions, respectively, we investigate their effects on those solutions, where x is the scaled propagation variable. When $\alpha$
 (x), $\beta$
 (x), $\gamma$
 (x) and $\delta$
 (x) are chosen as the linear, parabolic and periodic functions, we obtain the parabolic, cubic and quasi-periodic solitons, respectively. Head-on and overtaking interactions between the two solitons are presented, and the interactions are elastic. Besides, with certain values of the spectral parameter $\lambda$
 , a shock region between the two solitons appears, and the interaction is inelastic. Interactions between two kinds of the breathers are also studied, and we find that the interaction regions are similar to those of the second-order rogue waves. Rogue waves are split into some first-order rogue waves when $\alpha$
 (x), $\beta$
 (x), $\gamma$
 (x) and $\delta$
 (x) are the periodic or odd-numbered functions.

Solitons, breathers and rogue waves for a sixth-order variable-coefficient nonlinear Schrödinger equation in an ocean or optical fiber

In this paper, a fifth-order variable-coefficient nonlinear Schrodinger equation is investigated, which describes the propagation of the attosecond pulses in an optical fiber. Via the Hirota’s method and auxiliary functions, bilinear forms and dark one-, two- and three-soliton solutions are obtained. Propagation and interaction of the solitons are discussed graphically: We observe that the solitonic velocities are only related to β1(x), β2(x), β3(x) and β4(x), the coefficients of the second-, third-, fourth- and fifth-order terms, respectively, with x being the scaled distance, while the solitonic amplitudes are related to β1(x), β2(x), β3(x), β4(x) as well as the wave number. When β1(x), β2(x), β3(x) and β4(x) are the constants, or the linear, quadratic and trigonometric functions of x, we obtain the linear, parabolic, cubic and periodic dark solitons, respectively. Interactions between (among) the two (three) solitons are depicted, which can be regarded to be elastic because the solitonic amplitudes remain unchanged except for some phase shifts after each interaction in an optical fiber.

Zhong-Zhou Lan

Papers

Solitons, breathers and rogue waves for a sixth-order variable-coefficient nonlinear Schrödinger equation in an ocean or optical fiber

Bilinear forms and dark-soliton solutions for a fifth-order variable-coefficient nonlinear Schrödinger equation in an optical fiber