[Paper Review] Solutions of Non-Integrable Equations by the Hirota Direct Method

Aslı Pekcan|ArXiv.org|Mar 30, 2006

Nonlinear Waves and Solitons8 references3 citations

TL;DR

This paper demonstrates that the Hirota direct method—typically used for integrable nonlinear PDEs—can also generate exact multi-soliton solutions for non-integrable extensions of the KP and Boussinesq equations. By deriving bilinear and Hirota bilinear forms, the authors establish three- and four-Hirota solution conditions (3HC and 4HC), enabling exact soliton-like solutions even when integrability conditions (e.g., $a = b^2/12$ for eKP) are not satisfied.

ABSTRACT

We show that we can also apply the Hirota method to some non-integrable equations. For this purpose, we consider the extensions of the Kadomtsev-Petviashvili (KP) and the Boussinesq (Bo) equations. We present several solutions of these equations.

Motivation & Objective

• To extend the applicability of the Hirota direct method beyond integrable equations to non-integrable nonlinear PDEs.
• To investigate whether exact multi-soliton solutions can be constructed for extended KP and Boussinesq equations that lack full integrability.
• To derive and analyze the three-Hirota solution condition (3HC) and four-Hirota solution condition (4HC) for these non-integrable systems.
• To provide explicit classes of solutions under specific parameter constraints, including cases where $k_i = 0$ for two waves.
• To demonstrate the existence of soliton-like behavior in non-integrable settings through analytical and graphical solutions.

Proposed method

• Transform the extended KP (eKP) and Boussinesq (eBo) equations into bilinear forms using dependent variable transformations.
• Express the bilinear forms in terms of Hirota D-operators, enabling the construction of polynomial expressions in the D-operator (Hirota bilinear forms).
• Apply finite perturbation expansions in powers of a small parameter $\varepsilon$ to the solution ansatz $f = 1 + \varepsilon f_1 + \varepsilon^2 f_2 + \cdots$.
• Derive the three-Hirota solution condition (3HC) by requiring the coefficient of $\varepsilon^3$ to vanish, leading to a system of algebraic constraints on wave parameters.
• Derive the four-Hirota solution condition (4HC) by analyzing the $\varepsilon^4$ and $\varepsilon^5$ coefficient equations, yielding a consistency condition involving all four-wave interactions.
• Use dispersion relations to determine wave parameters $l_i$ from $k_i$, $w_i$, and constants $a$, $b$, ensuring consistency of the solution ansatz.

Experimental results

Research questions

• RQ1 Can the Hirota direct method be successfully applied to non-integrable extensions of the KP and Boussinesq equations?
• RQ2 What algebraic conditions must be satisfied for three- and four-soliton solutions to exist in non-integrable systems?
• RQ3 Are there specific parameter regimes—such as when two wave numbers are zero—where the 3HC and 4HC conditions are automatically satisfied?
• RQ4 Do the resulting solutions exhibit soliton-like behavior even when the equations are not integrable?
• RQ5 How do the derived solution conditions (3HC and 4HC) relate to known integrability criteria for the original KP and Boussinesq equations?

Key findings

• The Hirota direct method successfully produces exact one-, two-, three-, and four-soliton solutions for the non-integrable extended KP (eKP) and extended Boussinesq (eBo) equations.
• The three-Hirota solution condition (3HC) is derived as a necessary algebraic constraint on wave parameters $k_i$, $w_i$, $l_i$, and is satisfied in specific cases such as when any two $k_i = 0$.
• The four-Hirota solution condition (4HC) is formulated as a symmetric sum over all four-wave combinations: $\sum_{\sigma_i=\pm1} P(\sum \sigma_i p_i) \prod_{i<j} P(\sigma_i p_i - \sigma_j p_j) = 0$, which holds automatically in Case 1 (two $k_i = 0$).
• For the eBo equation with $a=5$, $b=8$, and $k_1=k_2=0$, $k_3=2$, $k_4=3$, the four-soliton solution is explicitly constructed and graphically visualized across time steps from $t=-6$ to $t=6$.
• The solution for the eKP equation with $a=5$, $b=8$, $k_1=1$, $k_2=2$, $w_1=-3$, $w_2=-5$ is also explicitly derived and visualized, showing soliton-like propagation despite non-integrability.
• The consistency between the $\varepsilon^4$ and $\varepsilon^5$ solution coefficients leads to the condition $C = A(1,2)A(1,3)A(1,4)A(2,3)A(2,4)A(3,4)$, confirming the validity of the four-soliton ansatz under the 4HC.